| L(s) = 1 | + (1.62 + 0.609i)3-s + 1.44·5-s + 3.46i·7-s + (2.25 + 1.97i)9-s − 6.00i·11-s − i·13-s + (2.33 + 0.878i)15-s + 4.92i·17-s + 1.73·19-s + (−2.11 + 5.62i)21-s + 4.37·23-s − 2.92·25-s + (2.45 + 4.57i)27-s − 0.607·29-s + 8.83i·31-s + ⋯ |
| L(s) = 1 | + (0.936 + 0.351i)3-s + 0.644·5-s + 1.31i·7-s + (0.752 + 0.658i)9-s − 1.81i·11-s − 0.277i·13-s + (0.603 + 0.226i)15-s + 1.19i·17-s + 0.398·19-s + (−0.461 + 1.22i)21-s + 0.912·23-s − 0.584·25-s + (0.472 + 0.881i)27-s − 0.112·29-s + 1.58i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.015090426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.015090426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.62 - 0.609i)T \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 6.00iT - 11T^{2} \) |
| 17 | \( 1 - 4.92iT - 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 + 0.607T + 29T^{2} \) |
| 31 | \( 1 - 8.83iT - 31T^{2} \) |
| 37 | \( 1 - 3.88iT - 37T^{2} \) |
| 41 | \( 1 + 7.61iT - 41T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 6.64iT - 61T^{2} \) |
| 67 | \( 1 + 5.83T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 + 0.230iT - 79T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 - 0.471iT - 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.808376329224134696800031763780, −8.638365075837695098323543142284, −7.78511208757365463226521626599, −6.65592416900329732037941162708, −5.60115372536887376182560624167, −5.49091451562725602586191106076, −4.02836664033089157463748392366, −3.13476350813081857792263888398, −2.52447683449900597860138185352, −1.38396934291891671936862822682,
0.977394140745723089103988651998, 2.03449163026419945395127612289, 2.82824857548935656601393720316, 4.14531710953164249455856298619, 4.48488461276300818979381324649, 5.75660560466001275654341106932, 6.89202474481715149417660205155, 7.38215254911109009219734140616, 7.67016888258599781509548761069, 9.052268501829083001538159343878
