| L(s) = 1 | + 4·2-s + 2·3-s + 10·4-s − 2·5-s + 8·6-s + 2·7-s + 20·8-s + 9-s − 8·10-s + 2·11-s + 20·12-s − 4·13-s + 8·14-s − 4·15-s + 35·16-s + 4·17-s + 4·18-s + 12·19-s − 20·20-s + 4·21-s + 8·22-s + 8·23-s + 40·24-s + 25-s − 16·26-s − 2·27-s + 20·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 5·4-s − 0.894·5-s + 3.26·6-s + 0.755·7-s + 7.07·8-s + 1/3·9-s − 2.52·10-s + 0.603·11-s + 5.77·12-s − 1.10·13-s + 2.13·14-s − 1.03·15-s + 35/4·16-s + 0.970·17-s + 0.942·18-s + 2.75·19-s − 4.47·20-s + 0.872·21-s + 1.70·22-s + 1.66·23-s + 8.16·24-s + 1/5·25-s − 3.13·26-s − 0.384·27-s + 3.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(44.53090305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(44.53090305\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - p T^{2} + 14 T^{3} + 60 T^{4} + 14 p T^{5} - p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_4\times C_2$ | \( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 112 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 46 T^{2} + 435 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T - 6 T^{2} - 128 T^{3} + 299 T^{4} - 128 p T^{5} - 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 14 T + 73 T^{2} - 238 T^{3} + 1932 T^{4} - 238 p T^{5} + 73 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 83 T^{2} - 6 T^{3} + 8556 T^{4} - 6 p T^{5} - 83 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 6 T^{2} + 496 T^{3} - 6013 T^{4} + 496 p T^{5} - 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 107 T^{2} + 138 T^{3} + 10572 T^{4} + 138 p T^{5} - 107 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.25644729152691080445132938237, −6.88098410537339448176642206375, −6.69363605504370887591560494262, −6.64199057049906784210940044231, −6.42342064341223383446788770427, −5.77387490738977905225721319888, −5.71611330976889986228471935790, −5.35835415038875783852424626624, −5.33803651742809362580953128191, −5.00333511722406565367847323251, −4.86980095929789332818456272714, −4.81511315984825902205024287999, −4.43457742587136057054502253708, −3.90041585808802891451286500433, −3.78088894861765260221186342736, −3.74390266352252071840969878313, −3.30466956022544491499451502130, −3.12495196303928649145075565161, −3.00594743725426921109884920505, −2.78999455642625949447812630077, −2.28418655424084891227949085317, −1.92066459610761922666650533171, −1.76661990927838286419989684950, −1.09188113873534649490456895264, −0.923206742438276791233764286658,
0.923206742438276791233764286658, 1.09188113873534649490456895264, 1.76661990927838286419989684950, 1.92066459610761922666650533171, 2.28418655424084891227949085317, 2.78999455642625949447812630077, 3.00594743725426921109884920505, 3.12495196303928649145075565161, 3.30466956022544491499451502130, 3.74390266352252071840969878313, 3.78088894861765260221186342736, 3.90041585808802891451286500433, 4.43457742587136057054502253708, 4.81511315984825902205024287999, 4.86980095929789332818456272714, 5.00333511722406565367847323251, 5.33803651742809362580953128191, 5.35835415038875783852424626624, 5.71611330976889986228471935790, 5.77387490738977905225721319888, 6.42342064341223383446788770427, 6.64199057049906784210940044231, 6.69363605504370887591560494262, 6.88098410537339448176642206375, 7.25644729152691080445132938237
