| L(s) = 1 | − 3-s + 2.11i·5-s + i·7-s + 9-s − 3.80i·11-s + (0.854 + 3.50i)13-s − 2.11i·15-s + 6.38·17-s − 3.24i·19-s − i·21-s + 9.17·23-s + 0.535·25-s − 27-s − 10.3·29-s + 2.01i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.944i·5-s + 0.377i·7-s + 0.333·9-s − 1.14i·11-s + (0.236 + 0.971i)13-s − 0.545i·15-s + 1.54·17-s − 0.744i·19-s − 0.218i·21-s + 1.91·23-s + 0.107·25-s − 0.192·27-s − 1.93·29-s + 0.362i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677881355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677881355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-0.854 - 3.50i)T \) |
| good | 5 | \( 1 - 2.11iT - 5T^{2} \) |
| 11 | \( 1 + 3.80iT - 11T^{2} \) |
| 17 | \( 1 - 6.38T + 17T^{2} \) |
| 19 | \( 1 + 3.24iT - 19T^{2} \) |
| 23 | \( 1 - 9.17T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 2.01iT - 31T^{2} \) |
| 37 | \( 1 + 2.15iT - 37T^{2} \) |
| 41 | \( 1 + 7.40iT - 41T^{2} \) |
| 43 | \( 1 - 1.29T + 43T^{2} \) |
| 47 | \( 1 + 1.12iT - 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 15.9iT - 67T^{2} \) |
| 71 | \( 1 - 2.35iT - 71T^{2} \) |
| 73 | \( 1 - 0.690iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 - 16.1iT - 89T^{2} \) |
| 97 | \( 1 + 15.5iT - 97T^{2} \) |
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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
−8.480499312807900608232813536543, −7.34121955296591843890345207373, −7.02992860681392914272266321288, −6.15226206571563358353525993568, −5.55385432064939691183499127304, −4.84920244681076925631560848390, −3.55215447321539705119130377760, −3.17620816346270272181929755149, −1.97473754530438484671899792868, −0.72544773759919103853115234676,
0.874730061263843361877109104911, 1.51409378809102503469499761096, 2.97923486112513358962720117705, 3.90568992093949352808761137386, 4.75541162477224487958257483093, 5.37855706791572460045355527422, 5.88544310195587308300812431905, 7.08746219876953268960716938948, 7.51335436869359907857850781751, 8.257581746867877835915237239187
