| L(s) = 1 | + (−1.90 + 1.17i)5-s − 3.55i·7-s − 5.36i·11-s − 5.98i·13-s + 2.77·17-s − 6.96i·19-s − 1.07i·23-s + (2.25 − 4.46i)25-s + (5.10 + 1.70i)29-s + 1.76i·31-s + (4.16 + 6.77i)35-s − 9.65·37-s − 6.14i·41-s − 6.11·43-s + 10.2·47-s + ⋯ |
| L(s) = 1 | + (−0.851 + 0.524i)5-s − 1.34i·7-s − 1.61i·11-s − 1.65i·13-s + 0.673·17-s − 1.59i·19-s − 0.223i·23-s + (0.450 − 0.892i)25-s + (0.948 + 0.317i)29-s + 0.317i·31-s + (0.704 + 1.14i)35-s − 1.58·37-s − 0.959i·41-s − 0.932·43-s + 1.49·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.239181532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239181532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.90 - 1.17i)T \) |
| 29 | \( 1 + (-5.10 - 1.70i)T \) |
| good | 7 | \( 1 + 3.55iT - 7T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 + 5.98iT - 13T^{2} \) |
| 17 | \( 1 - 2.77T + 17T^{2} \) |
| 19 | \( 1 + 6.96iT - 19T^{2} \) |
| 23 | \( 1 + 1.07iT - 23T^{2} \) |
| 31 | \( 1 - 1.76iT - 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 + 6.14iT - 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 3.34iT - 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 11.7iT - 61T^{2} \) |
| 67 | \( 1 - 0.517iT - 67T^{2} \) |
| 71 | \( 1 - 2.84T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 6.96iT - 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 - 13.3iT - 89T^{2} \) |
| 97 | \( 1 + 2.54T + 97T^{2} \) |
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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
−7.86364133528684013331699797417, −7.18063263171869459955035553608, −6.65391640827089406807314568528, −5.67056546014295668982436899597, −4.93417802055662304507727286822, −3.96142249597846163586509378014, −3.25855600224596188966853434118, −2.86282176397831718008732319818, −0.911691057762486154273705207667, −0.42053470256387440560756857832,
1.55898361906359674651140802418, 2.11894542499428418699359531102, 3.36585513864275196144474236577, 4.19040549685535532726946730663, 4.83315182315744634267054188252, 5.52206406642496367314870632339, 6.47289666317760319119873945123, 7.10764870556792100734326892496, 7.986084536263201435872468475320, 8.403953012753762955641269948277
