L(s) = 1 | + 0.138·2-s − 3-s − 1.98·4-s − 5-s − 0.138·6-s + 5.07·7-s − 0.551·8-s + 9-s − 0.138·10-s − 5.07·11-s + 1.98·12-s − 3.67·13-s + 0.703·14-s + 15-s + 3.88·16-s − 2.60·17-s + 0.138·18-s − 6.74·19-s + 1.98·20-s − 5.07·21-s − 0.703·22-s − 3.21·23-s + 0.551·24-s + 25-s − 0.508·26-s − 27-s − 10.0·28-s + ⋯ |
L(s) = 1 | + 0.0979·2-s − 0.577·3-s − 0.990·4-s − 0.447·5-s − 0.0565·6-s + 1.91·7-s − 0.195·8-s + 0.333·9-s − 0.0438·10-s − 1.53·11-s + 0.571·12-s − 1.01·13-s + 0.188·14-s + 0.258·15-s + 0.971·16-s − 0.632·17-s + 0.0326·18-s − 1.54·19-s + 0.442·20-s − 1.10·21-s − 0.150·22-s − 0.669·23-s + 0.112·24-s + 0.200·25-s − 0.0997·26-s − 0.192·27-s − 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 0.138T + 2T^{2} \) |
| 7 | \( 1 - 5.07T + 7T^{2} \) |
| 11 | \( 1 + 5.07T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 31 | \( 1 + 0.787T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 + 2.61T + 43T^{2} \) |
| 47 | \( 1 + 1.11T + 47T^{2} \) |
| 53 | \( 1 - 0.619T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 8.90T + 61T^{2} \) |
| 67 | \( 1 + 0.524T + 67T^{2} \) |
| 71 | \( 1 - 0.195T + 71T^{2} \) |
| 73 | \( 1 - 1.13T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 18.1T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−10.71036717257401462349638848585, −10.04878451733879850790709130175, −8.481334370762593863836516670310, −8.195822582563093419338897438413, −7.13525267508195215340102311921, −5.43442329951346623055725652593, −4.91064908322104063662949674689, −4.15831804220320238566162843221, −2.12071642513684236974011304291, 0,
2.12071642513684236974011304291, 4.15831804220320238566162843221, 4.91064908322104063662949674689, 5.43442329951346623055725652593, 7.13525267508195215340102311921, 8.195822582563093419338897438413, 8.481334370762593863836516670310, 10.04878451733879850790709130175, 10.71036717257401462349638848585