| L(s) = 1 | − 11.1·5-s − 47.9i·7-s + 55.4i·11-s − 28.1·13-s − 71.2·17-s − 138. i·19-s + 181. i·23-s + 125.·25-s + 1.56e3·29-s + 404. i·31-s + 536. i·35-s − 1.19e3·37-s + 2.26e3·41-s − 2.88e3i·43-s − 972. i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.978i·7-s + 0.458i·11-s − 0.166·13-s − 0.246·17-s − 0.383i·19-s + 0.343i·23-s + 0.200·25-s + 1.86·29-s + 0.420i·31-s + 0.437i·35-s − 0.869·37-s + 1.35·41-s − 1.55i·43-s − 0.440i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.028070727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028070727\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 11.1T \) |
| good | 7 | \( 1 + 47.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 55.4iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 28.1T + 2.85e4T^{2} \) |
| 17 | \( 1 + 71.2T + 8.35e4T^{2} \) |
| 19 | \( 1 + 138. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 181. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.56e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 404. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.19e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.26e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.88e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 972. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.69e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.25e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.43e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 933. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 8.62e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.19e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 361. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.53e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 2.66e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.03e4T + 8.85e7T^{2} \) |
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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
−8.659469047097298141490932246792, −7.77935780472264986201231868000, −7.10822294229659761133267870048, −6.45977189712525377673279392983, −5.17594125877274722409048949791, −4.42026938458932174580066265105, −3.62767156312555818845183271391, −2.53338960237605568779682462473, −1.23291438066332300467691116516, −0.23340622136197472061978724038,
1.02443906492191162551416183564, 2.36106132383720178832774891816, 3.14301937916016215753652781450, 4.28111043542974754936124172566, 5.12782471175310753857279340866, 6.07500833579934513284297149522, 6.73394013111051363932363351838, 7.936092949412477063854328972531, 8.379339358093249562748688111630, 9.233628173299017348225257365441
