| L(s) = 1 | − 2-s + 3.23·3-s + 4-s − 5-s − 3.23·6-s − 2.61·7-s − 8-s + 7.47·9-s + 10-s + 3.23·12-s + 1.61·13-s + 2.61·14-s − 3.23·15-s + 16-s + 5.23·17-s − 7.47·18-s + 4.09·19-s − 20-s − 8.47·21-s + 0.145·23-s − 3.23·24-s + 25-s − 1.61·26-s + 14.4·27-s − 2.61·28-s + 1.23·29-s + 3.23·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.86·3-s + 0.5·4-s − 0.447·5-s − 1.32·6-s − 0.989·7-s − 0.353·8-s + 2.49·9-s + 0.316·10-s + 0.934·12-s + 0.448·13-s + 0.699·14-s − 0.835·15-s + 0.250·16-s + 1.26·17-s − 1.76·18-s + 0.938·19-s − 0.223·20-s − 1.84·21-s + 0.0304·23-s − 0.660·24-s + 0.200·25-s − 0.317·26-s + 2.78·27-s − 0.494·28-s + 0.229·29-s + 0.590·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.119302360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119302360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 - 0.145T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 0.854T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 43 | \( 1 - 9.23T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 0.909T + 53T^{2} \) |
| 59 | \( 1 - 2.38T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 1.70T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.485546546344515658303031623540, −9.020117914018887128852309870897, −8.154282609039941413717392264995, −7.51903847205411447776543352266, −6.95337259877675577187907734752, −5.66494433196878283870052222158, −3.99980373744748919123389781390, −3.34326090201067843302575779225, −2.60632031089986872035062694993, −1.20878608321599233149146862544,
1.20878608321599233149146862544, 2.60632031089986872035062694993, 3.34326090201067843302575779225, 3.99980373744748919123389781390, 5.66494433196878283870052222158, 6.95337259877675577187907734752, 7.51903847205411447776543352266, 8.154282609039941413717392264995, 9.020117914018887128852309870897, 9.485546546344515658303031623540
