| L(s) = 1 | + 2-s − 2·3-s + 4-s − 4·5-s − 2·6-s + 2·7-s − 2·8-s + 7·9-s − 4·10-s + 4·11-s − 2·12-s + 2·14-s + 8·15-s − 8·17-s + 7·18-s + 4·19-s − 4·20-s − 4·21-s + 4·22-s + 6·23-s + 4·24-s + 10·25-s − 22·27-s + 2·28-s − 2·29-s + 8·30-s − 16·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s − 0.816·6-s + 0.755·7-s − 0.707·8-s + 7/3·9-s − 1.26·10-s + 1.20·11-s − 0.577·12-s + 0.534·14-s + 2.06·15-s − 1.94·17-s + 1.64·18-s + 0.917·19-s − 0.894·20-s − 0.872·21-s + 0.852·22-s + 1.25·23-s + 0.816·24-s + 2·25-s − 4.23·27-s + 0.377·28-s − 0.371·29-s + 1.46·30-s − 2.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5884263140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5884263140\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T + 3 T^{3} - 5 T^{4} + 3 p T^{5} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 3 T^{2} + 36 T^{3} - 128 T^{4} + 36 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 27 T^{2} + 24 T^{3} - 8 T^{4} + 24 p T^{5} + 27 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 13 T^{2} + 36 T^{3} + 176 T^{4} + 36 p T^{5} - 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 102 T^{3} - 908 T^{4} - 102 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 61 T^{2} + 2352 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 7 T^{2} + 6 p T^{3} - 36 p T^{4} + 6 p^{2} T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{2} - 3480 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 83 T^{2} + 972 T^{3} - 11544 T^{4} + 972 p T^{5} + 83 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 2 T - 123 T^{2} - 102 T^{3} + 7852 T^{4} - 102 p T^{5} - 123 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 24 T + 251 T^{2} - 3144 T^{3} + 40344 T^{4} - 3144 p T^{5} + 251 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−11.38273753470751407937937588343, −10.87905492944601137255752967700, −10.83936797101497974980434560023, −10.42582258176282826104586338613, −10.09638830681397758136885600406, −9.520704082235653297044018337427, −9.078392531186914942929991916857, −9.036532808057901851766936685973, −8.850316335612125910152289476152, −8.308002667946383299678719688113, −7.52540775289160939623334917265, −7.46498094166597590559225299668, −7.36207989365010637180779909424, −6.87919402287606897317926620573, −6.78905538776817669651575731269, −6.18456918179466007260633386603, −5.57902814091702612338913124920, −5.39734069479177377929684220852, −5.18213621752467731457333050423, −4.36800531468517690847863369810, −3.97776970866971482582701838201, −3.95016738525998569579110113141, −3.63455688525051791274289816000, −2.52759809345947439612794892322, −1.51545986592200855846070605594,
1.51545986592200855846070605594, 2.52759809345947439612794892322, 3.63455688525051791274289816000, 3.95016738525998569579110113141, 3.97776970866971482582701838201, 4.36800531468517690847863369810, 5.18213621752467731457333050423, 5.39734069479177377929684220852, 5.57902814091702612338913124920, 6.18456918179466007260633386603, 6.78905538776817669651575731269, 6.87919402287606897317926620573, 7.36207989365010637180779909424, 7.46498094166597590559225299668, 7.52540775289160939623334917265, 8.308002667946383299678719688113, 8.850316335612125910152289476152, 9.036532808057901851766936685973, 9.078392531186914942929991916857, 9.520704082235653297044018337427, 10.09638830681397758136885600406, 10.42582258176282826104586338613, 10.83936797101497974980434560023, 10.87905492944601137255752967700, 11.38273753470751407937937588343
