| L(s) = 1 | − 4·5-s − 4·9-s + 16·13-s − 2·16-s + 4·17-s − 8·19-s + 8·25-s + 12·29-s − 8·31-s − 4·37-s + 12·41-s + 16·45-s + 16·49-s + 40·53-s − 8·59-s − 64·65-s + 32·67-s − 12·71-s + 20·73-s + 24·79-s + 8·80-s + 10·81-s + 44·83-s − 16·85-s + 16·89-s + 32·95-s − 8·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 4/3·9-s + 4.43·13-s − 1/2·16-s + 0.970·17-s − 1.83·19-s + 8/5·25-s + 2.22·29-s − 1.43·31-s − 0.657·37-s + 1.87·41-s + 2.38·45-s + 16/7·49-s + 5.49·53-s − 1.04·59-s − 7.93·65-s + 3.90·67-s − 1.42·71-s + 2.34·73-s + 2.70·79-s + 0.894·80-s + 10/9·81-s + 4.82·83-s − 1.73·85-s + 1.69·89-s + 3.28·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.374579485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.374579485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 7 | \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | \( 1 - 16 T + 114 T^{2} - 512 T^{3} + 1890 T^{4} - 512 p T^{5} + 114 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 5 | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 41 T^{4} + 128 T^{5} + 312 T^{6} + 828 T^{7} + 2176 T^{8} + 828 p T^{9} + 312 p^{2} T^{10} + 128 p^{3} T^{11} + 41 p^{4} T^{12} + 16 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 48 T^{3} + 57 T^{4} + 768 T^{5} + 1152 T^{6} + 1872 T^{7} - 34240 T^{8} + 1872 p T^{9} + 1152 p^{2} T^{10} + 768 p^{3} T^{11} + 57 p^{4} T^{12} - 48 p^{5} T^{13} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 T + 45 T^{2} - 134 T^{3} + 944 T^{4} - 134 p T^{5} + 45 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 8 T + 32 T^{2} + 176 T^{3} + 51 p T^{4} + 3200 T^{5} + 10080 T^{6} + 36696 T^{7} + 117184 T^{8} + 36696 p T^{9} + 10080 p^{2} T^{10} + 3200 p^{3} T^{11} + 51 p^{5} T^{12} + 176 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 38 T^{2} + 1333 T^{4} - 32994 T^{6} + 926696 T^{8} - 32994 p^{2} T^{10} + 1333 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 6 T + 71 T^{2} - 298 T^{3} + 2534 T^{4} - 298 p T^{5} + 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 8 T + 32 T^{2} + 216 T^{3} + 2372 T^{4} + 14392 T^{5} + 62560 T^{6} + 474792 T^{7} + 3604870 T^{8} + 474792 p T^{9} + 62560 p^{2} T^{10} + 14392 p^{3} T^{11} + 2372 p^{4} T^{12} + 216 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 + 4 T + 8 T^{2} + 112 T^{3} + 3465 T^{4} + 11872 T^{5} + 26040 T^{6} + 394524 T^{7} + 5896384 T^{8} + 394524 p T^{9} + 26040 p^{2} T^{10} + 11872 p^{3} T^{11} + 3465 p^{4} T^{12} + 112 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 - 12 T + 72 T^{2} - 436 T^{3} + 2176 T^{4} - 11692 T^{5} + 78680 T^{6} - 488660 T^{7} + 3032766 T^{8} - 488660 p T^{9} + 78680 p^{2} T^{10} - 11692 p^{3} T^{11} + 2176 p^{4} T^{12} - 436 p^{5} T^{13} + 72 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 6 p T^{2} + 31169 T^{4} - 2332954 T^{6} + 119560116 T^{8} - 2332954 p^{2} T^{10} + 31169 p^{4} T^{12} - 6 p^{7} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 3682 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 20 T + 278 T^{2} - 2892 T^{3} + 23978 T^{4} - 2892 p T^{5} + 278 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 8 T + 32 T^{2} - 248 T^{3} + 2076 T^{4} - 7864 T^{5} - 98592 T^{6} - 1411448 T^{7} + 8207270 T^{8} - 1411448 p T^{9} - 98592 p^{2} T^{10} - 7864 p^{3} T^{11} + 2076 p^{4} T^{12} - 248 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 14 T^{2} + 7685 T^{4} + 105546 T^{6} + 30490872 T^{8} + 105546 p^{2} T^{10} + 7685 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 32 T + 512 T^{2} - 5600 T^{3} + 50684 T^{4} - 415584 T^{5} + 3028480 T^{6} - 18496928 T^{7} + 122079910 T^{8} - 18496928 p T^{9} + 3028480 p^{2} T^{10} - 415584 p^{3} T^{11} + 50684 p^{4} T^{12} - 5600 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 12 T + 72 T^{2} + 908 T^{3} + 2064 T^{4} - 51444 T^{5} - 353704 T^{6} - 5033588 T^{7} - 69761186 T^{8} - 5033588 p T^{9} - 353704 p^{2} T^{10} - 51444 p^{3} T^{11} + 2064 p^{4} T^{12} + 908 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 - 20 T + 200 T^{2} - 576 T^{3} - 9079 T^{4} + 107216 T^{5} - 162632 T^{6} - 7946652 T^{7} + 106585744 T^{8} - 7946652 p T^{9} - 162632 p^{2} T^{10} + 107216 p^{3} T^{11} - 9079 p^{4} T^{12} - 576 p^{5} T^{13} + 200 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 12 T + 246 T^{2} - 1836 T^{3} + 318 p T^{4} - 1836 p T^{5} + 246 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 44 T + 968 T^{2} - 16060 T^{3} + 235728 T^{4} - 3037100 T^{5} + 34409496 T^{6} - 357330556 T^{7} + 3416693630 T^{8} - 357330556 p T^{9} + 34409496 p^{2} T^{10} - 3037100 p^{3} T^{11} + 235728 p^{4} T^{12} - 16060 p^{5} T^{13} + 968 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( 1 - 16 T + 128 T^{2} - 1328 T^{3} + 23068 T^{4} - 260752 T^{5} + 2101120 T^{6} - 20928816 T^{7} + 206877318 T^{8} - 20928816 p T^{9} + 2101120 p^{2} T^{10} - 260752 p^{3} T^{11} + 23068 p^{4} T^{12} - 1328 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 + 8 T + 32 T^{2} + 1032 T^{3} - 988 T^{4} - 101096 T^{5} - 244640 T^{6} - 6736104 T^{7} - 156517562 T^{8} - 6736104 p T^{9} - 244640 p^{2} T^{10} - 101096 p^{3} T^{11} - 988 p^{4} T^{12} + 1032 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.69007545242350126944335696187, −4.65709499604539760677481861391, −4.54277261530459335503779521909, −4.12882491463326027231597007463, −4.08307034644370563219510960333, −3.96732833121641968829494732540, −3.74967385998910336813199788179, −3.68449735809379613504525882092, −3.56501021195138620335728771643, −3.55946022736346593788122226841, −3.48552596847837310251339022203, −3.33343598364815961319364089047, −3.13983751760370926054878140267, −2.72202364009245272597172537559, −2.59274395589200428529471229923, −2.33912737336825666681579191485, −2.28439175297976300577720139861, −2.21894489055507114865926331312, −1.95472897536364406063810010244, −1.85674640290608875164837534144, −1.09664939771924606348323149099, −1.08502958334600674661382483690, −0.844475772182072899889591142894, −0.76481628727888641746598696797, −0.55118220855173462400731389571,
0.55118220855173462400731389571, 0.76481628727888641746598696797, 0.844475772182072899889591142894, 1.08502958334600674661382483690, 1.09664939771924606348323149099, 1.85674640290608875164837534144, 1.95472897536364406063810010244, 2.21894489055507114865926331312, 2.28439175297976300577720139861, 2.33912737336825666681579191485, 2.59274395589200428529471229923, 2.72202364009245272597172537559, 3.13983751760370926054878140267, 3.33343598364815961319364089047, 3.48552596847837310251339022203, 3.55946022736346593788122226841, 3.56501021195138620335728771643, 3.68449735809379613504525882092, 3.74967385998910336813199788179, 3.96732833121641968829494732540, 4.08307034644370563219510960333, 4.12882491463326027231597007463, 4.54277261530459335503779521909, 4.65709499604539760677481861391, 4.69007545242350126944335696187
Plot not available for L-functions of degree greater than 10.
