| L(s) = 1 | + 1.37·2-s − 0.115·4-s − 5-s + 2.43·7-s − 2.90·8-s − 1.37·10-s + 11-s − 4.84·13-s + 3.33·14-s − 3.75·16-s + 1.04·17-s + 19-s + 0.115·20-s + 1.37·22-s − 0.377·23-s + 25-s − 6.65·26-s − 0.280·28-s − 4.50·29-s + 4.76·31-s + 0.652·32-s + 1.42·34-s − 2.43·35-s + 3.01·37-s + 1.37·38-s + 2.90·40-s − 0.790·41-s + ⋯ |
| L(s) = 1 | + 0.970·2-s − 0.0577·4-s − 0.447·5-s + 0.919·7-s − 1.02·8-s − 0.434·10-s + 0.301·11-s − 1.34·13-s + 0.892·14-s − 0.938·16-s + 0.252·17-s + 0.229·19-s + 0.0258·20-s + 0.292·22-s − 0.0786·23-s + 0.200·25-s − 1.30·26-s − 0.0530·28-s − 0.836·29-s + 0.855·31-s + 0.115·32-s + 0.244·34-s − 0.411·35-s + 0.495·37-s + 0.222·38-s + 0.459·40-s − 0.123·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.448964360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.448964360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.37T + 2T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 23 | \( 1 + 0.377T + 23T^{2} \) |
| 29 | \( 1 + 4.50T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 + 0.790T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 1.32T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 + 1.20T + 67T^{2} \) |
| 71 | \( 1 - 0.259T + 71T^{2} \) |
| 73 | \( 1 + 6.27T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 1.23T + 89T^{2} \) |
| 97 | \( 1 - 2.95T + 97T^{2} \) |
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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
−7.55530371889594673122209421457, −7.08978661494359634020602907383, −6.11105420553103396448195782259, −5.46189366462132249871088025623, −4.78004656688654531759655878877, −4.40802364004294928505957247267, −3.59909220698155811660773212401, −2.81367407526385054507662147608, −1.94503333726236556977837009981, −0.63638443166995431433772475117,
0.63638443166995431433772475117, 1.94503333726236556977837009981, 2.81367407526385054507662147608, 3.59909220698155811660773212401, 4.40802364004294928505957247267, 4.78004656688654531759655878877, 5.46189366462132249871088025623, 6.11105420553103396448195782259, 7.08978661494359634020602907383, 7.55530371889594673122209421457
