| L(s) = 1 | + 0.273·2-s − 2.79·3-s − 1.92·4-s − 0.828·5-s − 0.763·6-s + 5.25·7-s − 1.07·8-s + 4.78·9-s − 0.226·10-s − 0.517·11-s + 5.37·12-s − 0.886·13-s + 1.43·14-s + 2.31·15-s + 3.55·16-s + 1.08·17-s + 1.31·18-s − 4.51·19-s + 1.59·20-s − 14.6·21-s − 0.141·22-s − 7.63·23-s + 2.99·24-s − 4.31·25-s − 0.242·26-s − 4.98·27-s − 10.1·28-s + ⋯ |
| L(s) = 1 | + 0.193·2-s − 1.61·3-s − 0.962·4-s − 0.370·5-s − 0.311·6-s + 1.98·7-s − 0.379·8-s + 1.59·9-s − 0.0716·10-s − 0.156·11-s + 1.55·12-s − 0.245·13-s + 0.384·14-s + 0.596·15-s + 0.889·16-s + 0.264·17-s + 0.308·18-s − 1.03·19-s + 0.356·20-s − 3.20·21-s − 0.0301·22-s − 1.59·23-s + 0.611·24-s − 0.862·25-s − 0.0475·26-s − 0.960·27-s − 1.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 619 | \( 1 + T \) |
| good | 2 | \( 1 - 0.273T + 2T^{2} \) |
| 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 + 0.828T + 5T^{2} \) |
| 7 | \( 1 - 5.25T + 7T^{2} \) |
| 11 | \( 1 + 0.517T + 11T^{2} \) |
| 13 | \( 1 + 0.886T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 + 4.51T + 19T^{2} \) |
| 23 | \( 1 + 7.63T + 23T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 3.78T + 37T^{2} \) |
| 41 | \( 1 - 3.25T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 + 9.97T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 7.03T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 3.78T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 - 6.06T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−10.55748090249186024314812008723, −9.444620171199767464803684113936, −8.132818538062163028914923817257, −7.76034444618584733573252001538, −6.21044130309351994085415577963, −5.42741980085901053493553356917, −4.66353437709144412754707032366, −4.11639713059864392048441147848, −1.64245245536142282989261354143, 0,
1.64245245536142282989261354143, 4.11639713059864392048441147848, 4.66353437709144412754707032366, 5.42741980085901053493553356917, 6.21044130309351994085415577963, 7.76034444618584733573252001538, 8.132818538062163028914923817257, 9.444620171199767464803684113936, 10.55748090249186024314812008723
