| L(s) = 1 | + (1.64 + 1.51i)5-s − 2.05·7-s − 5.60·11-s + 2.63i·13-s − 2.12i·17-s − 4.87i·19-s + (4.47 − 1.71i)23-s + (0.412 + 4.98i)25-s + 1.54i·29-s − 6.05·31-s + (−3.38 − 3.11i)35-s − 0.289·37-s + 2.93i·41-s + 12.2·43-s − 0.475·47-s + ⋯ |
| L(s) = 1 | + (0.735 + 0.677i)5-s − 0.777·7-s − 1.68·11-s + 0.729i·13-s − 0.516i·17-s − 1.11i·19-s + (0.933 − 0.358i)23-s + (0.0825 + 0.996i)25-s + 0.287i·29-s − 1.08·31-s + (−0.571 − 0.526i)35-s − 0.0475·37-s + 0.459i·41-s + 1.86·43-s − 0.0693·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024272985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024272985\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.64 - 1.51i)T \) |
| 23 | \( 1 + (-4.47 + 1.71i)T \) |
| good | 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 2.63iT - 13T^{2} \) |
| 17 | \( 1 + 2.12iT - 17T^{2} \) |
| 19 | \( 1 + 4.87iT - 19T^{2} \) |
| 29 | \( 1 - 1.54iT - 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 + 0.289T + 37T^{2} \) |
| 41 | \( 1 - 2.93iT - 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 0.475T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 + 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 0.760iT - 61T^{2} \) |
| 67 | \( 1 + 0.897T + 67T^{2} \) |
| 71 | \( 1 + 1.57iT - 71T^{2} \) |
| 73 | \( 1 - 3.69iT - 73T^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 6.38iT - 83T^{2} \) |
| 89 | \( 1 + 4.36T + 89T^{2} \) |
| 97 | \( 1 - 1.66T + 97T^{2} \) |
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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
−8.257228598239564186018042286711, −7.24928703730517402645207765026, −6.93528547715004301164245228147, −6.09257249771331087447345689294, −5.30193904372619255370729278739, −4.67980258732082176423122129064, −3.33100343074613524821986693332, −2.76696127878018988673182413087, −2.00170344133301045358769957808, −0.31062851512974209817563663687,
1.02867675469759121956614720834, 2.26860658396929468381185792991, 3.00535982583481709404424515675, 4.00030429763902959144300758048, 5.04346723089000933031225694656, 5.71084300882565021806649853723, 6.02875886113288912648619754020, 7.28200906867129660791829615747, 7.81456469963461128996526543794, 8.618380832537287583303687751655
