| L(s) = 1 | − 3.65·2-s − 3.03·3-s + 5.35·4-s − 5·5-s + 11.1·6-s + 26.1·7-s + 9.67·8-s − 17.7·9-s + 18.2·10-s + 11·11-s − 16.2·12-s + 54.3·13-s − 95.6·14-s + 15.1·15-s − 78.1·16-s + 65.9·17-s + 64.9·18-s + 19·19-s − 26.7·20-s − 79.4·21-s − 40.1·22-s + 198.·23-s − 29.3·24-s + 25·25-s − 198.·26-s + 136.·27-s + 140.·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 0.584·3-s + 0.669·4-s − 0.447·5-s + 0.755·6-s + 1.41·7-s + 0.427·8-s − 0.658·9-s + 0.577·10-s + 0.301·11-s − 0.391·12-s + 1.16·13-s − 1.82·14-s + 0.261·15-s − 1.22·16-s + 0.941·17-s + 0.850·18-s + 0.229·19-s − 0.299·20-s − 0.826·21-s − 0.389·22-s + 1.80·23-s − 0.249·24-s + 0.200·25-s − 1.49·26-s + 0.969·27-s + 0.945·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.059385119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059385119\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 3.65T + 8T^{2} \) |
| 3 | \( 1 + 3.03T + 27T^{2} \) |
| 7 | \( 1 - 26.1T + 343T^{2} \) |
| 13 | \( 1 - 54.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 198.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 76.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 71.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 168.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 392.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 186.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 526.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 530.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 45.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 514.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 753.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 956.T + 9.12e5T^{2} \) |
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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
−9.409211888021848121655280018575, −8.426183047165831624999248370520, −8.288789799937332343041866206452, −7.30840932975815186290214235683, −6.33090139731759126710832978660, −5.19337107525411134613594480557, −4.47502674004360040242986968501, −3.01658491881316845713969611701, −1.36359957890397968752640449141, −0.809500326864412004610818955388,
0.809500326864412004610818955388, 1.36359957890397968752640449141, 3.01658491881316845713969611701, 4.47502674004360040242986968501, 5.19337107525411134613594480557, 6.33090139731759126710832978660, 7.30840932975815186290214235683, 8.288789799937332343041866206452, 8.426183047165831624999248370520, 9.409211888021848121655280018575
