| L(s) = 1 | + (−1.83 − 1.27i)5-s + 0.920·7-s − 4.88·11-s + 0.822i·13-s − 5.93i·17-s − 7.43i·19-s + (2.12 + 4.29i)23-s + (1.75 + 4.68i)25-s − 2.63i·29-s + 5.24·31-s + (−1.69 − 1.17i)35-s + 7.23·37-s + 1.93i·41-s − 1.42·43-s − 13.2·47-s + ⋯ |
| L(s) = 1 | + (−0.821 − 0.569i)5-s + 0.347·7-s − 1.47·11-s + 0.228i·13-s − 1.43i·17-s − 1.70i·19-s + (0.443 + 0.896i)23-s + (0.350 + 0.936i)25-s − 0.488i·29-s + 0.941·31-s + (−0.285 − 0.198i)35-s + 1.18·37-s + 0.301i·41-s − 0.217·43-s − 1.93·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02539715727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02539715727\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.83 + 1.27i)T \) |
| 23 | \( 1 + (-2.12 - 4.29i)T \) |
| good | 7 | \( 1 - 0.920T + 7T^{2} \) |
| 11 | \( 1 + 4.88T + 11T^{2} \) |
| 13 | \( 1 - 0.822iT - 13T^{2} \) |
| 17 | \( 1 + 5.93iT - 17T^{2} \) |
| 19 | \( 1 + 7.43iT - 19T^{2} \) |
| 29 | \( 1 + 2.63iT - 29T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 - 1.93iT - 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 0.0569iT - 53T^{2} \) |
| 59 | \( 1 - 2.50iT - 59T^{2} \) |
| 61 | \( 1 - 5.95iT - 61T^{2} \) |
| 67 | \( 1 + 5.00T + 67T^{2} \) |
| 71 | \( 1 - 8.61iT - 71T^{2} \) |
| 73 | \( 1 + 3.83iT - 73T^{2} \) |
| 79 | \( 1 - 5.42iT - 79T^{2} \) |
| 83 | \( 1 + 1.52iT - 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.560884374153530159704741462815, −7.931146375272513056558616433366, −7.39220566251467272888577773771, −6.68963610020184232526312545870, −5.45410375456347704229062250925, −4.87101567801084711265038031904, −4.44553389067153409885128101963, −3.11402030512265954593245328011, −2.55913385916909100028765819318, −1.05290909676999748820638636584,
0.008129922820390541558510179221, 1.59516757158773193707924881912, 2.71817940544233158487060311923, 3.45010579353857712657737410689, 4.32608998350522985122687032943, 5.08559924616130334719965918581, 6.02673808730206916661042715033, 6.63109602264148097531422008381, 7.69311142795658519634471805649, 8.142797307936906027012270602293
