| L(s) = 1 | − 2·2-s − 4-s + 4·5-s + 10·7-s + 4·8-s − 8·10-s − 20·14-s + 2·17-s + 16·19-s − 4·20-s + 10·23-s + 10·25-s − 10·28-s − 8·29-s + 8·31-s − 2·32-s − 4·34-s + 40·35-s − 2·37-s − 32·38-s + 16·40-s − 8·41-s − 2·43-s − 20·46-s − 8·47-s + 42·49-s − 20·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 1.78·5-s + 3.77·7-s + 1.41·8-s − 2.52·10-s − 5.34·14-s + 0.485·17-s + 3.67·19-s − 0.894·20-s + 2.08·23-s + 2·25-s − 1.88·28-s − 1.48·29-s + 1.43·31-s − 0.353·32-s − 0.685·34-s + 6.76·35-s − 0.328·37-s − 5.19·38-s + 2.52·40-s − 1.24·41-s − 0.304·43-s − 2.94·46-s − 1.16·47-s + 6·49-s − 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.32981862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.32981862\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + 5 T^{2} + p^{3} T^{3} + 13 T^{4} + p^{4} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 58 T^{2} - 232 T^{3} + 703 T^{4} - 232 p T^{5} + 58 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 14 T^{2} + 9 p T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T + 50 T^{2} - 92 T^{3} + 1135 T^{4} - 92 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 8 T + 51 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 98 T^{2} - 544 T^{3} + 137 p T^{4} - 544 p T^{5} + 98 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 98 T^{2} + 656 T^{3} + 4003 T^{4} + 656 p T^{5} + 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 94 T^{2} + 260 T^{3} + 4219 T^{4} + 260 p T^{5} + 94 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 83 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 154 T^{2} + 248 T^{3} + 9559 T^{4} + 248 p T^{5} + 154 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 116 T^{2} + 392 T^{3} + 5158 T^{4} + 392 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 248 T^{2} - 36 p T^{3} + 20622 T^{4} - 36 p^{2} T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 266 T^{2} + 2136 T^{3} + 24423 T^{4} + 2136 p T^{5} + 266 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 28 T + 502 T^{2} - 6088 T^{3} + 55063 T^{4} - 6088 p T^{5} + 502 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 30 T + 598 T^{2} - 7608 T^{3} + 73923 T^{4} - 7608 p T^{5} + 598 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 74 T^{2} + 424 T^{3} + 10903 T^{4} + 424 p T^{5} + 74 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 208 T^{2} + 920 T^{3} + 17998 T^{4} + 920 p T^{5} + 208 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 308 T^{2} - 2700 T^{3} + 37158 T^{4} - 2700 p T^{5} + 308 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 122 T^{2} - 1056 T^{3} + 14727 T^{4} - 1056 p T^{5} + 122 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 298 T^{2} - 956 T^{3} + 38551 T^{4} - 956 p T^{5} + 298 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.42379855949288328868716993606, −5.34126500718100647118747827161, −5.14498642940450894347672429682, −5.09134396865681225531407185640, −5.02181359694962127006887578588, −4.63857679301806920129171115337, −4.50182766069631201092513944311, −4.36525307011471871736990959294, −4.29065238667945428273172559728, −3.71495975255576731253001988924, −3.53327570218927946350942303765, −3.44007545219917676071486130728, −3.18070481855639347135578920699, −2.97423489740894986468448335840, −2.80283788392211156741054225868, −2.39736940936600332719783038790, −2.19953957514865454891235764994, −1.92059528366334706453358521179, −1.64732387994604418231175904511, −1.62016166265135182658613918251, −1.51316113151363091522723459848, −0.979446968289519755625232916340, −0.847599902754695506243994549040, −0.68759074507928419909804053108, −0.64218813108593808816518722240,
0.64218813108593808816518722240, 0.68759074507928419909804053108, 0.847599902754695506243994549040, 0.979446968289519755625232916340, 1.51316113151363091522723459848, 1.62016166265135182658613918251, 1.64732387994604418231175904511, 1.92059528366334706453358521179, 2.19953957514865454891235764994, 2.39736940936600332719783038790, 2.80283788392211156741054225868, 2.97423489740894986468448335840, 3.18070481855639347135578920699, 3.44007545219917676071486130728, 3.53327570218927946350942303765, 3.71495975255576731253001988924, 4.29065238667945428273172559728, 4.36525307011471871736990959294, 4.50182766069631201092513944311, 4.63857679301806920129171115337, 5.02181359694962127006887578588, 5.09134396865681225531407185640, 5.14498642940450894347672429682, 5.34126500718100647118747827161, 5.42379855949288328868716993606
