L(s) = 1 | + 4·2-s + 6·4-s + 16·11-s − 15·16-s + 64·22-s − 16·23-s − 8·25-s − 16·29-s − 24·32-s − 16·37-s − 8·43-s + 96·44-s − 64·46-s − 32·50-s + 16·53-s − 64·58-s − 6·64-s − 36·67-s − 48·71-s − 64·74-s − 8·79-s − 32·86-s − 96·92-s − 48·100-s + 64·106-s − 28·107-s + 8·109-s + ⋯ |
L(s) = 1 | + 2.82·2-s + 3·4-s + 4.82·11-s − 3.75·16-s + 13.6·22-s − 3.33·23-s − 8/5·25-s − 2.97·29-s − 4.24·32-s − 2.63·37-s − 1.21·43-s + 14.4·44-s − 9.43·46-s − 4.52·50-s + 2.19·53-s − 8.40·58-s − 3/4·64-s − 4.39·67-s − 5.69·71-s − 7.43·74-s − 0.900·79-s − 3.45·86-s − 10.0·92-s − 4.79·100-s + 6.21·106-s − 2.70·107-s + 0.766·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.875363153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.875363153\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( ( 1 + 4 T^{2} + 6 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 4 T^{2} + 106 T^{4} + 1712 T^{6} - 23165 T^{8} + 1712 p^{2} T^{10} + 106 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 8 T^{2} - 455 T^{4} - 472 T^{6} + 167344 T^{8} - 472 p^{2} T^{10} - 455 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 48 T^{2} + 1033 T^{4} - 26352 T^{6} + 653376 T^{8} - 26352 p^{2} T^{10} + 1033 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( 1 - 76 T^{2} + 2842 T^{4} - 76912 T^{6} + 2134099 T^{8} - 76912 p^{2} T^{10} + 2842 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 8 T - 14 T^{2} + 32 T^{3} + 2347 T^{4} + 32 p T^{5} - 14 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 88 T^{2} + 3769 T^{4} - 53944 T^{6} - 42800 T^{8} - 53944 p^{2} T^{10} + 3769 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 4 T - 71 T^{2} + 4 T^{3} + 5032 T^{4} + 4 p T^{5} - 71 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 76 T^{2} + 2986 T^{4} + 123728 T^{6} - 9854765 T^{8} + 123728 p^{2} T^{10} + 2986 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 8 T - 10 T^{2} + 256 T^{3} - 725 T^{4} + 256 p T^{5} - 10 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 160 T^{2} + 13561 T^{4} - 812320 T^{6} + 43706560 T^{8} - 812320 p^{2} T^{10} + 13561 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 36 T^{2} - 3398 T^{4} + 98928 T^{6} + 3365379 T^{8} + 98928 p^{2} T^{10} - 3398 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 18 T + 121 T^{2} + 1242 T^{3} + 15012 T^{4} + 1242 p T^{5} + 121 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 216 T^{2} + 24409 T^{4} - 2503224 T^{6} + 217900944 T^{8} - 2503224 p^{2} T^{10} + 24409 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 4 T - 98 T^{2} - 176 T^{3} + 5491 T^{4} - 176 p T^{5} - 98 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 32 T^{2} + 6190 T^{4} - 606208 T^{6} - 27825101 T^{8} - 606208 p^{2} T^{10} + 6190 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 24 T^{2} + 3289 T^{4} + 516744 T^{6} - 86588496 T^{8} + 516744 p^{2} T^{10} + 3289 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.77270947372665121411073484189, −3.72105259286060591901793402241, −3.63625533970583814828468624824, −3.56982817720446920863602739155, −3.49654225242110558210866939024, −3.05000108132516899128148312269, −2.98668187692287378261444117146, −2.83894326564009810589623182077, −2.83765577357234685836726861634, −2.82851346356519400829729360265, −2.82617742494612125003750554497, −2.25080183264088994708685889755, −2.20790403648055201097741526190, −1.88207067806001700022014619818, −1.87817423891224964845696791548, −1.75867476307007818278235439593, −1.75168986250497225936125577746, −1.65058224693382158802347277447, −1.43589135710035617476275694921, −1.42619831889698812285876070632, −1.05823230756900400081178821362, −0.979983422690407229856585845511, −0.40475952531485826816144046482, −0.34770590512463103051771986481, −0.084740466078501274848349560385,
0.084740466078501274848349560385, 0.34770590512463103051771986481, 0.40475952531485826816144046482, 0.979983422690407229856585845511, 1.05823230756900400081178821362, 1.42619831889698812285876070632, 1.43589135710035617476275694921, 1.65058224693382158802347277447, 1.75168986250497225936125577746, 1.75867476307007818278235439593, 1.87817423891224964845696791548, 1.88207067806001700022014619818, 2.20790403648055201097741526190, 2.25080183264088994708685889755, 2.82617742494612125003750554497, 2.82851346356519400829729360265, 2.83765577357234685836726861634, 2.83894326564009810589623182077, 2.98668187692287378261444117146, 3.05000108132516899128148312269, 3.49654225242110558210866939024, 3.56982817720446920863602739155, 3.63625533970583814828468624824, 3.72105259286060591901793402241, 3.77270947372665121411073484189
Plot not available for L-functions of degree greater than 10.