| L(s) = 1 | − 4·4-s + 4·5-s − 4·9-s + 10·16-s − 24·19-s − 16·20-s + 10·25-s + 16·29-s + 32·31-s + 16·36-s + 24·41-s − 16·45-s − 40·59-s + 24·61-s − 20·64-s − 40·71-s + 96·76-s + 16·79-s + 40·80-s + 10·81-s − 88·89-s − 96·95-s − 40·100-s + 48·101-s − 24·109-s − 64·116-s − 52·121-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.78·5-s − 4/3·9-s + 5/2·16-s − 5.50·19-s − 3.57·20-s + 2·25-s + 2.97·29-s + 5.74·31-s + 8/3·36-s + 3.74·41-s − 2.38·45-s − 5.20·59-s + 3.07·61-s − 5/2·64-s − 4.74·71-s + 11.0·76-s + 1.80·79-s + 4.47·80-s + 10/9·81-s − 9.32·89-s − 9.84·95-s − 4·100-s + 4.77·101-s − 2.29·109-s − 5.94·116-s − 4.72·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4177618022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4177618022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| 5 | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 11 | \( ( 1 + 26 T^{2} - 16 T^{3} + 362 T^{4} - 16 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 4 p T^{2} + 1320 T^{4} - 23228 T^{6} + 329390 T^{8} - 23228 p^{2} T^{10} + 1320 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 56 T^{2} + 1268 T^{4} - 11176 T^{6} + 45670 T^{8} - 11176 p^{2} T^{10} + 1268 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 12 T + 88 T^{2} + 460 T^{3} + 2174 T^{4} + 460 p T^{5} + 88 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 96 T^{2} + 4220 T^{4} - 116640 T^{6} + 2693446 T^{8} - 116640 p^{2} T^{10} + 4220 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 8 T + 50 T^{2} - 360 T^{3} + 2786 T^{4} - 360 p T^{5} + 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 16 T + 202 T^{2} - 1616 T^{3} + 10666 T^{4} - 1616 p T^{5} + 202 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 4 p T^{2} + 9560 T^{4} - 370620 T^{6} + 12535566 T^{8} - 370620 p^{2} T^{10} + 9560 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 12 T + 190 T^{2} - 1356 T^{3} + 11842 T^{4} - 1356 p T^{5} + 190 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 108 T^{2} + 7400 T^{4} - 450340 T^{6} + 22241646 T^{8} - 450340 p^{2} T^{10} + 7400 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 172 T^{2} + 15464 T^{4} - 948164 T^{6} + 47811790 T^{8} - 948164 p^{2} T^{10} + 15464 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 240 T^{2} + 25532 T^{4} - 1715600 T^{6} + 94098918 T^{8} - 1715600 p^{2} T^{10} + 25532 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 20 T + 288 T^{2} + 2804 T^{3} + 24014 T^{4} + 2804 p T^{5} + 288 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 118 T^{2} - 684 T^{3} + 6306 T^{4} - 684 p T^{5} + 118 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 492 T^{2} + 108392 T^{4} - 13982820 T^{6} + 1156204078 T^{8} - 13982820 p^{2} T^{10} + 108392 p^{4} T^{12} - 492 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 20 T + 248 T^{2} + 2788 T^{3} + 28078 T^{4} + 2788 p T^{5} + 248 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 268 T^{2} + 30568 T^{4} - 2051940 T^{6} + 125567630 T^{8} - 2051940 p^{2} T^{10} + 30568 p^{4} T^{12} - 268 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 8 T + 140 T^{2} - 104 T^{3} + 6054 T^{4} - 104 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 424 T^{2} + 80348 T^{4} - 9509080 T^{6} + 861447654 T^{8} - 9509080 p^{2} T^{10} + 80348 p^{4} T^{12} - 424 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 44 T + 1014 T^{2} + 15436 T^{3} + 169698 T^{4} + 15436 p T^{5} + 1014 p^{2} T^{6} + 44 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 348 T^{2} + 81160 T^{4} - 12144532 T^{6} + 1394593550 T^{8} - 12144532 p^{2} T^{10} + 81160 p^{4} T^{12} - 348 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.14142726978107382616959459329, −4.09349286404083913769938130011, −4.00808344906820659418692543248, −3.82500222971617153509863546480, −3.32537994755828334772995216910, −3.23791243839408735456391988555, −3.12262590042375763855506076473, −3.09763064380491924083325948108, −3.04917030068978204318999567876, −2.91316819571186806665311956821, −2.61889768955266914783535128808, −2.55523549639596507158647859901, −2.45248716681999555701346070026, −2.34196292055494742292842965491, −2.19405761644086089522953361220, −2.10684397514615877681710765312, −2.04487790930792894240492300162, −1.43568227392360777057977869079, −1.29731367126722137171792257794, −1.26463438058089183687511714733, −1.25980443879619904709610678984, −0.889614785385598890267374279983, −0.817165491894354347360609516565, −0.22392332452715152657293592922, −0.14142492717513663951394777249,
0.14142492717513663951394777249, 0.22392332452715152657293592922, 0.817165491894354347360609516565, 0.889614785385598890267374279983, 1.25980443879619904709610678984, 1.26463438058089183687511714733, 1.29731367126722137171792257794, 1.43568227392360777057977869079, 2.04487790930792894240492300162, 2.10684397514615877681710765312, 2.19405761644086089522953361220, 2.34196292055494742292842965491, 2.45248716681999555701346070026, 2.55523549639596507158647859901, 2.61889768955266914783535128808, 2.91316819571186806665311956821, 3.04917030068978204318999567876, 3.09763064380491924083325948108, 3.12262590042375763855506076473, 3.23791243839408735456391988555, 3.32537994755828334772995216910, 3.82500222971617153509863546480, 4.00808344906820659418692543248, 4.09349286404083913769938130011, 4.14142726978107382616959459329
Plot not available for L-functions of degree greater than 10.
