| L(s) = 1 | + 3.46i·5-s + 24.2·7-s + 48i·11-s + 41.5i·13-s − 54·17-s − 4i·19-s − 173.·23-s + 113·25-s − 162. i·29-s + 58.8·31-s + 84i·35-s + 325. i·37-s + 294·41-s − 188i·43-s − 505.·47-s + ⋯ |
| L(s) = 1 | + 0.309i·5-s + 1.30·7-s + 1.31i·11-s + 0.886i·13-s − 0.770·17-s − 0.0482i·19-s − 1.57·23-s + 0.904·25-s − 1.04i·29-s + 0.341·31-s + 0.405i·35-s + 1.44i·37-s + 1.11·41-s − 0.666i·43-s − 1.56·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.766525913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766525913\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.46iT - 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 11 | \( 1 - 48iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 41.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 173.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 58.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 325. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 294T + 6.89e4T^{2} \) |
| 43 | \( 1 + 188iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 505.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 744. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 252iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 90.0iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 6.92T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 720iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−10.55571057601455360670491018454, −9.784333945209611655757377855703, −8.760377090105483343436252163855, −7.87634974717564821451407938956, −7.07041787902609996027711067623, −6.09766040698839433911227097284, −4.68168202979267780797657257751, −4.29023293072434679065508450379, −2.44693025787220538262170362356, −1.55632438754891291247340403404,
0.51763904594110609012799638514, 1.82816391527957270050066060978, 3.24653178779226834919270950067, 4.49930217344175819805746519605, 5.38231592395590540146583323258, 6.27631782757155007668327494359, 7.64117403804036566338358723607, 8.326042858026561696575640692405, 8.923475928164163640289549888784, 10.22774060949936410850809291687
