| L(s) = 1 | − 4·3-s + 5-s − 7-s + 10·9-s − 4·11-s − 4·13-s − 4·15-s − 2·17-s − 2·19-s + 4·21-s + 23-s − 25-s − 20·27-s + 29-s − 24·31-s + 16·33-s − 35-s + 4·37-s + 16·39-s + 7·41-s − 3·43-s + 10·45-s − 32·47-s − 11·49-s + 8·51-s + 24·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.447·5-s − 0.377·7-s + 10/3·9-s − 1.20·11-s − 1.10·13-s − 1.03·15-s − 0.485·17-s − 0.458·19-s + 0.872·21-s + 0.208·23-s − 1/5·25-s − 3.84·27-s + 0.185·29-s − 4.31·31-s + 2.78·33-s − 0.169·35-s + 0.657·37-s + 2.56·39-s + 1.09·41-s − 0.457·43-s + 1.49·45-s − 4.66·47-s − 1.57·49-s + 1.12·51-s + 3.29·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - T + 2 T^{2} - 7 T^{3} - 2 T^{4} - 7 p T^{5} + 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 12 T^{2} + 29 T^{3} + 86 T^{4} + 29 p T^{5} + 12 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 52 T^{2} + 106 T^{3} + 1206 T^{4} + 106 p T^{5} + 52 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 T + 36 T^{2} + 34 T^{3} + 614 T^{4} + 34 p T^{5} + 36 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - T + 76 T^{2} - 77 T^{3} + 2454 T^{4} - 77 p T^{5} + 76 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 70 T^{2} - 59 T^{3} + 2886 T^{4} - 59 p T^{5} + 70 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 24 T + 292 T^{2} + 2412 T^{3} + 15174 T^{4} + 2412 p T^{5} + 292 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 4 T + 84 T^{2} - 140 T^{3} + 3398 T^{4} - 140 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 7 T + 10 T^{2} - 65 T^{3} + 2010 T^{4} - 65 p T^{5} + 10 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 3 T + 76 T^{2} + 675 T^{3} + 2802 T^{4} + 675 p T^{5} + 76 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + T + 176 T^{2} + 337 T^{3} + 13774 T^{4} + 337 p T^{5} + 176 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 7 T + 202 T^{2} - 901 T^{3} + 16618 T^{4} - 901 p T^{5} + 202 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 5 T + 112 T^{2} - 569 T^{3} + 8074 T^{4} - 569 p T^{5} + 112 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 18 T + 332 T^{2} + 3546 T^{3} + 36486 T^{4} + 3546 p T^{5} + 332 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + T - 18 T^{2} + 23 T^{3} + 9842 T^{4} + 23 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 22 T + 480 T^{2} + 5666 T^{3} + 63638 T^{4} + 5666 p T^{5} + 480 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 22 T + 292 T^{2} + 2438 T^{3} + 20742 T^{4} + 2438 p T^{5} + 292 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 22 T + 376 T^{2} + 4334 T^{3} + 46542 T^{4} + 4334 p T^{5} + 376 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 4 T + 324 T^{2} - 860 T^{3} + 43958 T^{4} - 860 p T^{5} + 324 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.05586567507588703999062223925, −5.58805565548896183386292568115, −5.57547273754440958769841486361, −5.46356203533902313929523917484, −5.42567626878550793646942890065, −5.03906702933994279310281610620, −4.93008516580065890179663147204, −4.69076762432965438741033887876, −4.66212273762110088978351166369, −4.38040053563062642991547789631, −4.13231054167291820952490080419, −3.95863820793646690746530035277, −3.86556476856332895615746718206, −3.36427469461901626698229438840, −3.34847769559933966851108514087, −3.06152484628195239368786027641, −2.93595361003841850332637186963, −2.53961673816063946534486983242, −2.16296435987162314805554172577, −2.13394164846692831386730514564, −2.04289724047551387210865041518, −1.49502691646600597848655233106, −1.45641723301485362132118641730, −1.20302493350737175781978503214, −0.77025078363994532691524173295, 0, 0, 0, 0,
0.77025078363994532691524173295, 1.20302493350737175781978503214, 1.45641723301485362132118641730, 1.49502691646600597848655233106, 2.04289724047551387210865041518, 2.13394164846692831386730514564, 2.16296435987162314805554172577, 2.53961673816063946534486983242, 2.93595361003841850332637186963, 3.06152484628195239368786027641, 3.34847769559933966851108514087, 3.36427469461901626698229438840, 3.86556476856332895615746718206, 3.95863820793646690746530035277, 4.13231054167291820952490080419, 4.38040053563062642991547789631, 4.66212273762110088978351166369, 4.69076762432965438741033887876, 4.93008516580065890179663147204, 5.03906702933994279310281610620, 5.42567626878550793646942890065, 5.46356203533902313929523917484, 5.57547273754440958769841486361, 5.58805565548896183386292568115, 6.05586567507588703999062223925
