| L(s) = 1 | + 0.872·2-s + 2.26·3-s − 1.23·4-s + 5-s + 1.97·6-s − 2.82·8-s + 2.13·9-s + 0.872·10-s − 5.40·11-s − 2.80·12-s + 5.49·13-s + 2.26·15-s + 0.0113·16-s + 1.44·17-s + 1.86·18-s − 4.90·19-s − 1.23·20-s − 4.71·22-s − 3.85·23-s − 6.40·24-s + 25-s + 4.79·26-s − 1.95·27-s + 7.76·29-s + 1.97·30-s − 5.05·31-s + 5.66·32-s + ⋯ |
| L(s) = 1 | + 0.617·2-s + 1.30·3-s − 0.619·4-s + 0.447·5-s + 0.807·6-s − 0.999·8-s + 0.712·9-s + 0.275·10-s − 1.62·11-s − 0.810·12-s + 1.52·13-s + 0.585·15-s + 0.00284·16-s + 0.351·17-s + 0.439·18-s − 1.12·19-s − 0.276·20-s − 1.00·22-s − 0.802·23-s − 1.30·24-s + 0.200·25-s + 0.940·26-s − 0.376·27-s + 1.44·29-s + 0.361·30-s − 0.908·31-s + 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 0.872T + 2T^{2} \) |
| 3 | \( 1 - 2.26T + 3T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 + 5.05T + 31T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 + 7.03T + 47T^{2} \) |
| 53 | \( 1 - 2.32T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 + 1.09T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 - 8.06T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.71103030792923316361781844082, −6.50454828935170029586000690641, −5.94905752508148533508708944816, −5.25348876122445386966609330605, −4.47701861356375416563054595918, −3.74632138123028648290661452914, −3.09652747646929333541511574938, −2.50864242340050056634851800767, −1.53852229542559514013131566127, 0,
1.53852229542559514013131566127, 2.50864242340050056634851800767, 3.09652747646929333541511574938, 3.74632138123028648290661452914, 4.47701861356375416563054595918, 5.25348876122445386966609330605, 5.94905752508148533508708944816, 6.50454828935170029586000690641, 7.71103030792923316361781844082
