| L(s) = 1 | − 6.85i·2-s − 31.0·4-s − 11.1i·5-s − 25.1·7-s + 102. i·8-s − 76.6·10-s − 129. i·11-s − 287.·13-s + 172. i·14-s + 209.·16-s − 400. i·17-s − 74.2·19-s + 346. i·20-s − 886.·22-s + 166. i·23-s + ⋯ |
| L(s) = 1 | − 1.71i·2-s − 1.93·4-s − 0.447i·5-s − 0.512·7-s + 1.60i·8-s − 0.766·10-s − 1.06i·11-s − 1.70·13-s + 0.878i·14-s + 0.816·16-s − 1.38i·17-s − 0.205·19-s + 0.866i·20-s − 1.83·22-s + 0.313i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.01185422199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01185422199\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 11.1iT \) |
| good | 2 | \( 1 + 6.85iT - 16T^{2} \) |
| 7 | \( 1 + 25.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 129. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 287.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 400. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 74.2T + 1.30e5T^{2} \) |
| 23 | \( 1 - 166. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.12e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.05e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.50e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.21e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.33e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.49e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.03e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 602. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.73e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 47.0T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.80e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.27e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.64e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.44e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 4.96e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 3.06e3T + 8.85e7T^{2} \) |
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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
−10.75102264021520828225411314617, −9.698906293545436864471549504737, −9.357480994763436059494561085632, −8.248460001277317237961500851410, −6.93994606310168807641184158982, −5.35815095909818009210942144266, −4.51175108305167869975971482734, −3.21358227434443834006354756497, −2.48118632720795058168694756801, −0.949230786462536676408579231587,
0.00403367940161698408936228389, 2.38225125952027443164410195361, 4.13954626083191001167806408612, 4.98977220636757100885004182346, 6.20463897859010728487717646190, 6.80040433679385788818752358050, 7.67344200713861770881342196580, 8.401560792606163163539638144742, 9.854242829758224944685491103056, 9.939279260519306653340866718591
