| L(s) = 1 | + 2-s + 4·3-s − 4-s + 4·6-s − 4·7-s − 3·8-s + 10·9-s − 2·11-s − 4·12-s − 4·13-s − 4·14-s − 4·16-s − 3·17-s + 10·18-s − 13·19-s − 16·21-s − 2·22-s + 2·23-s − 12·24-s − 4·26-s + 20·27-s + 4·28-s + 7·29-s − 19·31-s − 32-s − 8·33-s − 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.30·3-s − 1/2·4-s + 1.63·6-s − 1.51·7-s − 1.06·8-s + 10/3·9-s − 0.603·11-s − 1.15·12-s − 1.10·13-s − 1.06·14-s − 16-s − 0.727·17-s + 2.35·18-s − 2.98·19-s − 3.49·21-s − 0.426·22-s + 0.417·23-s − 2.44·24-s − 0.784·26-s + 3.84·27-s + 0.755·28-s + 1.29·29-s − 3.41·31-s − 0.176·32-s − 1.39·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + p T^{2} + 3 T^{4} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + p T^{2} + 28 T^{3} + 269 T^{4} + 28 p T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 3 T + 38 T^{2} + 203 T^{3} + 693 T^{4} + 203 p T^{5} + 38 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 13 T + 127 T^{2} + 796 T^{3} + 4111 T^{4} + 796 p T^{5} + 127 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 2 T + 44 T^{2} - 104 T^{3} + 1477 T^{4} - 104 p T^{5} + 44 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 7 T + 122 T^{2} - 589 T^{3} + 5401 T^{4} - 589 p T^{5} + 122 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 19 T + 190 T^{2} + 1283 T^{3} + 7415 T^{4} + 1283 p T^{5} + 190 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 2 T + 82 T^{2} - 23 T^{3} + 3131 T^{4} - 23 p T^{5} + 82 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 11 T + 104 T^{2} + 297 T^{3} + 2073 T^{4} + 297 p T^{5} + 104 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 17 T + 205 T^{2} - 1858 T^{3} + 14279 T^{4} - 1858 p T^{5} + 205 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 10 T + 167 T^{2} + 1372 T^{3} + 11281 T^{4} + 1372 p T^{5} + 167 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 15 T + 248 T^{2} - 2367 T^{3} + 20643 T^{4} - 2367 p T^{5} + 248 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + T + 86 T^{2} - 495 T^{3} + 3495 T^{4} - 495 p T^{5} + 86 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 8 T + 118 T^{2} - 141 T^{3} + 3867 T^{4} - 141 p T^{5} + 118 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 6 T + 55 T^{2} - 172 T^{3} + 4965 T^{4} - 172 p T^{5} + 55 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 14 T + 164 T^{2} + 1576 T^{3} + 11309 T^{4} + 1576 p T^{5} + 164 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 24 T + 454 T^{2} + 5583 T^{3} + 55635 T^{4} + 5583 p T^{5} + 454 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 19 T + 373 T^{2} + 4190 T^{3} + 46241 T^{4} + 4190 p T^{5} + 373 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 16 T + 314 T^{2} + 3903 T^{3} + 38427 T^{4} + 3903 p T^{5} + 314 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 36 T + 782 T^{2} + 11537 T^{3} + 126195 T^{4} + 11537 p T^{5} + 782 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 29 T + 628 T^{2} + 8861 T^{3} + 102763 T^{4} + 8861 p T^{5} + 628 p^{2} T^{6} + 29 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.98920804337948892497469055528, −5.59850773439043057403230555934, −5.57114051543409385001895505400, −5.44175641089188848064060440977, −5.36077754618772949400046579000, −4.90661844530449136250253520917, −4.62752992371480739851904344564, −4.47953865599777302364629987911, −4.36400018301154232996146381133, −4.15171716315713258901320435587, −4.12472115013960831330111457859, −4.05648215274056549559247007676, −3.64562669031886076832869304802, −3.30375677971777727311093101910, −3.27864131301414769684399807891, −3.20897490035104827743832277196, −2.74901876437424105001926223920, −2.62369460062548683255117341882, −2.59763553997078142699621607556, −2.23652178251255782447586666338, −2.22398210855133496523612312391, −1.89905332614150511032506408120, −1.60137312869881220483968555195, −1.24258168778152755529300898355, −1.08933024369710836498534256485, 0, 0, 0, 0,
1.08933024369710836498534256485, 1.24258168778152755529300898355, 1.60137312869881220483968555195, 1.89905332614150511032506408120, 2.22398210855133496523612312391, 2.23652178251255782447586666338, 2.59763553997078142699621607556, 2.62369460062548683255117341882, 2.74901876437424105001926223920, 3.20897490035104827743832277196, 3.27864131301414769684399807891, 3.30375677971777727311093101910, 3.64562669031886076832869304802, 4.05648215274056549559247007676, 4.12472115013960831330111457859, 4.15171716315713258901320435587, 4.36400018301154232996146381133, 4.47953865599777302364629987911, 4.62752992371480739851904344564, 4.90661844530449136250253520917, 5.36077754618772949400046579000, 5.44175641089188848064060440977, 5.57114051543409385001895505400, 5.59850773439043057403230555934, 5.98920804337948892497469055528
