| L(s) = 1 | + 39.2·5-s − 95.4i·7-s − 61.4i·11-s − 119.·13-s − 147.·17-s − 566. i·19-s + 735. i·23-s + 913.·25-s − 1.07e3·29-s − 967. i·31-s − 3.74e3i·35-s + 705.·37-s − 2.04e3·41-s − 1.91e3i·43-s + 1.96e3i·47-s + ⋯ |
| L(s) = 1 | + 1.56·5-s − 1.94i·7-s − 0.508i·11-s − 0.708·13-s − 0.511·17-s − 1.56i·19-s + 1.39i·23-s + 1.46·25-s − 1.27·29-s − 1.00i·31-s − 3.05i·35-s + 0.515·37-s − 1.21·41-s − 1.03i·43-s + 0.888i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.475623985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475623985\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 39.2T + 625T^{2} \) |
| 7 | \( 1 + 95.4iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 61.4iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 119.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 147.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 566. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 735. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.07e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 967. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 705.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.04e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.91e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.96e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.49e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.64e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.31e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.39e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 4.66e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.68e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 425. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 6.99e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.12e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 2.27e3T + 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.976591381865660319693078391357, −7.62209881723729068139762558370, −7.09454135328757253819807843861, −6.27177895706932979384788995753, −5.33188209964607193629609262626, −4.49059819594352493956720997422, −3.43722667934418784039904603952, −2.27083754264779879765252800083, −1.26547529544857791127491014332, −0.25534819573230944777788609854,
1.82196361673493678752401763551, 2.10127245722444816006113047503, 3.11114809169900610547920359172, 4.77617027520796426364599156369, 5.43630206737287688034864702855, 6.10007561841683680499577983167, 6.75821487430914971103060923004, 8.159088856764593397164458251855, 8.795172892131377976893242564390, 9.666803165748163434400067757768
