| L(s) = 1 | + 2.12·2-s + 3-s + 2.51·4-s + 2.12·6-s + 3.64·7-s + 1.09·8-s + 9-s + 2.51·12-s + 1.51·13-s + 7.73·14-s − 2.70·16-s + 1.15·17-s + 2.12·18-s − 2.60·19-s + 3.64·21-s + 5.73·23-s + 1.09·24-s + 3.21·26-s + 27-s + 9.15·28-s − 6.24·29-s + 5.51·31-s − 7.93·32-s + 2.45·34-s + 2.51·36-s − 0.454·37-s − 5.54·38-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.577·3-s + 1.25·4-s + 0.867·6-s + 1.37·7-s + 0.387·8-s + 0.333·9-s + 0.726·12-s + 0.420·13-s + 2.06·14-s − 0.676·16-s + 0.280·17-s + 0.500·18-s − 0.598·19-s + 0.794·21-s + 1.19·23-s + 0.223·24-s + 0.631·26-s + 0.192·27-s + 1.73·28-s − 1.16·29-s + 0.990·31-s − 1.40·32-s + 0.420·34-s + 0.419·36-s − 0.0747·37-s − 0.899·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.724367835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.724367835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 13 | \( 1 - 1.51T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 + 0.454T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.48T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 8.51T + 71T^{2} \) |
| 73 | \( 1 + 9.21T + 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 6.77T + 97T^{2} \) |
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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
−7.49475075317083817490051913796, −7.05809799011028904744750972618, −6.13421413649613026653464758475, −5.44812547212522015382316516323, −4.90253866841712723387862683318, −4.18545952845182959659214357810, −3.71092064854082877092638272587, −2.72956488628604900771944456427, −2.12613638385070312616862171116, −1.11156754632507947773457392790,
1.11156754632507947773457392790, 2.12613638385070312616862171116, 2.72956488628604900771944456427, 3.71092064854082877092638272587, 4.18545952845182959659214357810, 4.90253866841712723387862683318, 5.44812547212522015382316516323, 6.13421413649613026653464758475, 7.05809799011028904744750972618, 7.49475075317083817490051913796
