| L(s) = 1 | − 2·2-s + 4·3-s + 2·4-s + 6·5-s − 8·6-s − 4·8-s + 11·9-s − 12·10-s + 8·11-s + 8·12-s − 8·13-s + 24·15-s + 8·16-s + 6·17-s − 22·18-s + 8·19-s + 12·20-s − 16·22-s − 12·23-s − 16·24-s + 27·25-s + 16·26-s + 20·27-s − 8·29-s − 48·30-s + 4·31-s − 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 4-s + 2.68·5-s − 3.26·6-s − 1.41·8-s + 11/3·9-s − 3.79·10-s + 2.41·11-s + 2.30·12-s − 2.21·13-s + 6.19·15-s + 2·16-s + 1.45·17-s − 5.18·18-s + 1.83·19-s + 2.68·20-s − 3.41·22-s − 2.50·23-s − 3.26·24-s + 27/5·25-s + 3.13·26-s + 3.84·27-s − 1.48·29-s − 8.76·30-s + 0.718·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.410327763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.410327763\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 152 T^{3} + 532 T^{4} - 152 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 446 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 18 T^{3} + 60 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 168 T^{3} + 644 T^{4} - 168 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 105 T^{2} + 684 T^{3} + 3824 T^{4} + 684 p T^{5} + 105 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1918 T^{4} + 248 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 47 T^{2} - 4 T^{3} + 2512 T^{4} - 4 p T^{5} - 47 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3494 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2702 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 17 T^{2} - 396 T^{3} + 7152 T^{4} - 396 p T^{5} + 17 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2 T + 101 T^{2} + 226 T^{3} + 4912 T^{4} + 226 p T^{5} + 101 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 28 T + 365 T^{2} + 3028 T^{3} + 22252 T^{4} + 3028 p T^{5} + 365 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 14 T + 53 T^{2} + 606 T^{3} - 9088 T^{4} + 606 p T^{5} + 53 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 45 T^{2} - 972 T^{3} - 11812 T^{4} - 972 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 18 T + 277 T^{2} - 3042 T^{3} + 31116 T^{4} - 3042 p T^{5} + 277 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 109 T^{2} + 176 T^{3} - 4856 T^{4} + 176 p T^{5} + 109 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 2180 T^{3} + 23086 T^{4} + 2180 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.60898878754437264014159096880, −7.17885156374054541170796249287, −7.09650943760235910372527641220, −7.02305985859651419583805421228, −6.33602219830750707540536529737, −6.23357749547065714080937179082, −6.07789714706169639787911760611, −6.01696952301148763267579316068, −5.67457745809841939320222931246, −5.34602459493885227089863889648, −5.14612745677162014922467178533, −4.69365711174394830383176276688, −4.33690393102488350416328051477, −4.22402176016667495552435324982, −3.89204355563626706256641861058, −3.47388350998889436386185165551, −3.30812201542658934109753896121, −2.87820135743805597379527799709, −2.62761389072671296559661208128, −2.45812917961110656926034285831, −2.13811252350927101375421630830, −1.95947157386748524725362093811, −1.32674550436089727126058050581, −1.14801447569952847446289982447, −0.978271154293716595598603291334,
0.978271154293716595598603291334, 1.14801447569952847446289982447, 1.32674550436089727126058050581, 1.95947157386748524725362093811, 2.13811252350927101375421630830, 2.45812917961110656926034285831, 2.62761389072671296559661208128, 2.87820135743805597379527799709, 3.30812201542658934109753896121, 3.47388350998889436386185165551, 3.89204355563626706256641861058, 4.22402176016667495552435324982, 4.33690393102488350416328051477, 4.69365711174394830383176276688, 5.14612745677162014922467178533, 5.34602459493885227089863889648, 5.67457745809841939320222931246, 6.01696952301148763267579316068, 6.07789714706169639787911760611, 6.23357749547065714080937179082, 6.33602219830750707540536529737, 7.02305985859651419583805421228, 7.09650943760235910372527641220, 7.17885156374054541170796249287, 7.60898878754437264014159096880
