| L(s) = 1 | + 4·2-s − 2·3-s + 10·4-s + 5-s − 8·6-s − 7-s + 20·8-s − 3·9-s + 4·10-s − 2·11-s − 20·12-s − 4·14-s − 2·15-s + 35·16-s + 17-s − 12·18-s + 4·19-s + 10·20-s + 2·21-s − 8·22-s − 10·23-s − 40·24-s − 8·25-s + 3·27-s − 10·28-s − 29-s − 8·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 1.15·3-s + 5·4-s + 0.447·5-s − 3.26·6-s − 0.377·7-s + 7.07·8-s − 9-s + 1.26·10-s − 0.603·11-s − 5.77·12-s − 1.06·14-s − 0.516·15-s + 35/4·16-s + 0.242·17-s − 2.82·18-s + 0.917·19-s + 2.23·20-s + 0.436·21-s − 1.70·22-s − 2.08·23-s − 8.16·24-s − 8/5·25-s + 0.577·27-s − 1.88·28-s − 0.185·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 2 T + 7 T^{2} + 17 T^{3} + 26 T^{4} + 17 p T^{5} + 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - T + 9 T^{2} - T^{3} + 39 T^{4} - p T^{5} + 9 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + T + 20 T^{2} + 22 T^{3} + 190 T^{4} + 22 p T^{5} + 20 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 35 T^{2} + 49 T^{3} + 534 T^{4} + 49 p T^{5} + 35 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - T + 2 p T^{2} + 53 T^{3} + 506 T^{4} + 53 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 10 T + 123 T^{2} + 725 T^{3} + 4612 T^{4} + 725 p T^{5} + 123 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + T + 47 T^{2} + 63 T^{3} + 2173 T^{4} + 63 p T^{5} + 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 11 T + 162 T^{2} + 1068 T^{3} + 8130 T^{4} + 1068 p T^{5} + 162 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 9 T + 144 T^{2} - 803 T^{3} + 7526 T^{4} - 803 p T^{5} + 144 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 4 T + 139 T^{2} + 381 T^{3} + 7989 T^{4} + 381 p T^{5} + 139 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 15 T + 203 T^{2} + 1768 T^{3} + 13888 T^{4} + 1768 p T^{5} + 203 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 25 T + 6 p T^{2} + 1850 T^{3} + 10924 T^{4} + 1850 p T^{5} + 6 p^{3} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 8 T + 217 T^{2} + 1253 T^{3} + 17383 T^{4} + 1253 p T^{5} + 217 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 28 T + 459 T^{2} + 5089 T^{3} + 44334 T^{4} + 5089 p T^{5} + 459 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 15 T + 277 T^{2} + 2575 T^{3} + 26425 T^{4} + 2575 p T^{5} + 277 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 104 T^{2} - 952 T^{3} + 3790 T^{4} - 952 p T^{5} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - T + 149 T^{2} - 332 T^{3} + 11898 T^{4} - 332 p T^{5} + 149 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 6 T + 168 T^{2} + 220 T^{3} + 161 p T^{4} + 220 p T^{5} + 168 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 12 T + 103 T^{2} + 25 T^{3} - 1502 T^{4} + 25 p T^{5} + 103 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 17 T + 410 T^{2} - 4216 T^{3} + 53744 T^{4} - 4216 p T^{5} + 410 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 15 T + 256 T^{2} - 3130 T^{3} + 28476 T^{4} - 3130 p T^{5} + 256 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2 T + 88 T^{2} - 826 T^{3} + 7006 T^{4} - 826 p T^{5} + 88 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.07184201075914200327824306009, −5.65125438000720416555810859303, −5.52433445653315655887759039715, −5.49454812860914380316534170074, −5.35902689641016784601027580053, −5.04415478891461421115219031106, −4.91871003250731411787490675181, −4.88348673535142254615887151124, −4.54500785615240455034181389891, −4.49778003394046249847244588012, −4.06260026186455991304001736767, −3.86415083561560989116062934211, −3.84020811932445560650661570585, −3.33760632037321052098521401766, −3.33090274111033091331200899876, −3.26927557298093658610742421936, −3.19269853984445474460304803321, −2.67339365145238606064192970968, −2.59221633270879231798157302330, −2.39030651881079722483094841138, −1.86312003837836049025944120515, −1.85518282527335566798113766330, −1.59231687556036808421402654100, −1.40588890725304002389207978212, −1.32941731446558777073897928656, 0, 0, 0, 0,
1.32941731446558777073897928656, 1.40588890725304002389207978212, 1.59231687556036808421402654100, 1.85518282527335566798113766330, 1.86312003837836049025944120515, 2.39030651881079722483094841138, 2.59221633270879231798157302330, 2.67339365145238606064192970968, 3.19269853984445474460304803321, 3.26927557298093658610742421936, 3.33090274111033091331200899876, 3.33760632037321052098521401766, 3.84020811932445560650661570585, 3.86415083561560989116062934211, 4.06260026186455991304001736767, 4.49778003394046249847244588012, 4.54500785615240455034181389891, 4.88348673535142254615887151124, 4.91871003250731411787490675181, 5.04415478891461421115219031106, 5.35902689641016784601027580053, 5.49454812860914380316534170074, 5.52433445653315655887759039715, 5.65125438000720416555810859303, 6.07184201075914200327824306009
