| L(s) = 1 | − 4·3-s + 2·5-s + 2·7-s + 10·9-s + 4·11-s − 4·13-s − 8·15-s + 12·17-s + 4·19-s − 8·21-s + 2·23-s − 4·25-s − 20·27-s + 4·29-s − 16·33-s + 4·35-s + 8·37-s + 16·39-s + 22·41-s − 14·43-s + 20·45-s + 6·47-s − 4·49-s − 48·51-s + 10·53-s + 8·55-s − 16·57-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 0.755·7-s + 10/3·9-s + 1.20·11-s − 1.10·13-s − 2.06·15-s + 2.91·17-s + 0.917·19-s − 1.74·21-s + 0.417·23-s − 4/5·25-s − 3.84·27-s + 0.742·29-s − 2.78·33-s + 0.676·35-s + 1.31·37-s + 2.56·39-s + 3.43·41-s − 2.13·43-s + 2.98·45-s + 0.875·47-s − 4/7·49-s − 6.72·51-s + 1.37·53-s + 1.07·55-s − 2.11·57-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
| \(L(1)\) |
\(\approx\) |
\(8.171816653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.171816653\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2 \wr S_4$ | \( 1 - 2 T + 8 T^{2} - 4 T^{3} + 18 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 2 T + 8 T^{2} - 22 T^{3} + 110 T^{4} - 22 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 12 T + 110 T^{2} - 38 p T^{3} + 3130 T^{4} - 38 p^{2} T^{5} + 110 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T + 12 T^{2} - 88 T^{3} + 862 T^{4} - 88 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 2 T + 64 T^{2} - 110 T^{3} + 2086 T^{4} - 110 p T^{5} + 64 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 110 T^{2} - 326 T^{3} + 4686 T^{4} - 326 p T^{5} + 110 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 36 T^{2} + 194 T^{3} + 1010 T^{4} + 194 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 8 T + 124 T^{2} - 744 T^{3} + 6310 T^{4} - 744 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 22 T + 324 T^{2} - 3134 T^{3} + 23566 T^{4} - 3134 p T^{5} + 324 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 14 T + 146 T^{2} + 992 T^{3} + 6610 T^{4} + 992 p T^{5} + 146 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 6 T + 128 T^{2} - 598 T^{3} + 7774 T^{4} - 598 p T^{5} + 128 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 10 T + 144 T^{2} - 1054 T^{3} + 9582 T^{4} - 1054 p T^{5} + 144 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 4 T + 188 T^{2} - 500 T^{3} + 15030 T^{4} - 500 p T^{5} + 188 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 18 T + 156 T^{2} - 574 T^{3} + 1222 T^{4} - 574 p T^{5} + 156 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 22 T + 436 T^{2} + 4952 T^{3} + 50022 T^{4} + 4952 p T^{5} + 436 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 2 T + 192 T^{2} - 290 T^{3} + 17310 T^{4} - 290 p T^{5} + 192 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 16 T + 288 T^{2} - 2476 T^{3} + 27502 T^{4} - 2476 p T^{5} + 288 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 2 T + 78 T^{2} + 768 T^{3} + 5330 T^{4} + 768 p T^{5} + 78 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 8 T + 308 T^{2} + 1816 T^{3} + 37158 T^{4} + 1816 p T^{5} + 308 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 304 T^{2} + 1770 T^{3} + 38658 T^{4} + 1770 p T^{5} + 304 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 10 T + 276 T^{2} + 1374 T^{3} + 30470 T^{4} + 1374 p T^{5} + 276 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.64460715949381487976679621382, −5.41818277692942698151922719872, −5.38780595528413933116988975805, −5.26758173554411238040785678882, −4.83745142856720156176407228830, −4.66385898139838818222177072751, −4.57514844553501477633599870435, −4.38604704063916474386233552653, −4.30155632270538662410667120875, −3.82593835959396601067915724802, −3.69390255951432829737529816055, −3.68089092817863755786904417572, −3.42125786876437098523572995639, −2.91606031101325742703737455997, −2.73497933007082785852372636216, −2.69580645772928172283538649122, −2.48885804260417302665524269202, −1.82517732489455078019172746977, −1.75584941161371623992475653440, −1.70258175675835228022917360100, −1.45484001789329169635601964447, −0.833640142458285721345400699089, −0.831402941851401604634623007606, −0.76591396097624263838697747680, −0.49409254777573007464335669385,
0.49409254777573007464335669385, 0.76591396097624263838697747680, 0.831402941851401604634623007606, 0.833640142458285721345400699089, 1.45484001789329169635601964447, 1.70258175675835228022917360100, 1.75584941161371623992475653440, 1.82517732489455078019172746977, 2.48885804260417302665524269202, 2.69580645772928172283538649122, 2.73497933007082785852372636216, 2.91606031101325742703737455997, 3.42125786876437098523572995639, 3.68089092817863755786904417572, 3.69390255951432829737529816055, 3.82593835959396601067915724802, 4.30155632270538662410667120875, 4.38604704063916474386233552653, 4.57514844553501477633599870435, 4.66385898139838818222177072751, 4.83745142856720156176407228830, 5.26758173554411238040785678882, 5.38780595528413933116988975805, 5.41818277692942698151922719872, 5.64460715949381487976679621382
