| L(s) = 1 | + 4·2-s + 2·3-s + 11·4-s + 8·6-s + 22·8-s − 9-s + 2·11-s + 22·12-s − 8·13-s + 36·16-s − 4·17-s − 4·18-s + 2·19-s + 8·22-s + 4·23-s + 44·24-s − 32·26-s − 2·27-s − 12·31-s + 52·32-s + 4·33-s − 16·34-s − 11·36-s − 12·37-s + 8·38-s − 16·39-s + 6·43-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 11/2·4-s + 3.26·6-s + 7.77·8-s − 1/3·9-s + 0.603·11-s + 6.35·12-s − 2.21·13-s + 9·16-s − 0.970·17-s − 0.942·18-s + 0.458·19-s + 1.70·22-s + 0.834·23-s + 8.98·24-s − 6.27·26-s − 0.384·27-s − 2.15·31-s + 9.19·32-s + 0.696·33-s − 2.74·34-s − 1.83·36-s − 1.97·37-s + 1.29·38-s − 2.56·39-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.61685573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.61685573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T + 5 T^{2} + p T^{3} - 11 T^{4} + p^{2} T^{5} + 5 p^{2} T^{6} - p^{5} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T + 20 T^{2} + 100 T^{3} + 271 T^{4} + 100 p T^{5} + 20 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 4 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 244 p T^{5} + 53 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 106 T^{2} + 696 T^{3} + 3891 T^{4} + 696 p T^{5} + 106 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 60 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 528 p T^{5} + 90 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 108 p T^{5} - 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 1308 p T^{5} + 157 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 22 T + 137 T^{2} + 834 T^{3} - 16648 T^{4} + 834 p T^{5} + 137 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 144 T^{2} + 24 p T^{3} - 31057 T^{4} + 24 p^{2} T^{5} + 144 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 816 T^{3} + 13203 T^{4} - 816 p T^{5} + 148 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 140 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + 16 T + p T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 8818 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.500717930306257253046347502308, −9.046183783987994648384291707190, −8.569685585005744129840119747749, −8.505379268652946696695483118186, −8.313065062143812665050290758394, −7.72448275058583270428299641636, −7.34447332494334016284520953283, −7.32467885066155209930607391064, −7.12724396592915383925060826454, −6.60067732444547471466632222025, −6.36236822680905203260413741577, −6.34923318938250373679724253664, −5.94905406398640255453906854376, −5.24016235661242346419875234577, −5.12866380750197197337521256065, −4.98589100177498931811556459960, −4.84279489707216291796343415807, −4.23881748632353773422303780243, −3.79926247855083371229138324898, −3.25084719096874303214038866598, −3.24149175554138190633566813546, −3.15469266268573948495990219819, −2.26494140339108451262283559894, −2.19408184833160732483297456449, −1.91864997835047486560858897831,
1.91864997835047486560858897831, 2.19408184833160732483297456449, 2.26494140339108451262283559894, 3.15469266268573948495990219819, 3.24149175554138190633566813546, 3.25084719096874303214038866598, 3.79926247855083371229138324898, 4.23881748632353773422303780243, 4.84279489707216291796343415807, 4.98589100177498931811556459960, 5.12866380750197197337521256065, 5.24016235661242346419875234577, 5.94905406398640255453906854376, 6.34923318938250373679724253664, 6.36236822680905203260413741577, 6.60067732444547471466632222025, 7.12724396592915383925060826454, 7.32467885066155209930607391064, 7.34447332494334016284520953283, 7.72448275058583270428299641636, 8.313065062143812665050290758394, 8.505379268652946696695483118186, 8.569685585005744129840119747749, 9.046183783987994648384291707190, 9.500717930306257253046347502308
