| L(s) = 1 | − 1.53·3-s + 2.27·5-s + 5.10·7-s − 0.656·9-s + 1.89·11-s + 3.24·13-s − 3.48·15-s + 6.89·17-s − 19-s − 7.81·21-s + 1.52·23-s + 0.190·25-s + 5.59·27-s − 1.20·29-s + 6.43·31-s − 2.90·33-s + 11.6·35-s + 0.689·37-s − 4.96·39-s − 2.78·41-s + 0.850·43-s − 1.49·45-s − 2.45·47-s + 19.0·49-s − 10.5·51-s − 53-s + 4.31·55-s + ⋯ |
| L(s) = 1 | − 0.883·3-s + 1.01·5-s + 1.92·7-s − 0.218·9-s + 0.571·11-s + 0.899·13-s − 0.900·15-s + 1.67·17-s − 0.229·19-s − 1.70·21-s + 0.317·23-s + 0.0381·25-s + 1.07·27-s − 0.223·29-s + 1.15·31-s − 0.504·33-s + 1.96·35-s + 0.113·37-s − 0.795·39-s − 0.434·41-s + 0.129·43-s − 0.222·45-s − 0.358·47-s + 2.72·49-s − 1.47·51-s − 0.137·53-s + 0.582·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.536505797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536505797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 - 5.10T + 7T^{2} \) |
| 11 | \( 1 - 1.89T + 11T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 - 6.43T + 31T^{2} \) |
| 37 | \( 1 - 0.689T + 37T^{2} \) |
| 41 | \( 1 + 2.78T + 41T^{2} \) |
| 43 | \( 1 - 0.850T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 5.74T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 8.87T + 79T^{2} \) |
| 83 | \( 1 - 5.38T + 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
| 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
−8.387982879707162257043738685418, −7.83939355453179968057681846324, −6.82501020715183982917350035160, −5.93621481217910119677661422369, −5.58282838826882402879571041131, −4.92160580707567218444543087888, −4.07732420839011004881163231693, −2.82531947651429516884482281212, −1.59044695836001970675127076661, −1.12454212109580631300820199753,
1.12454212109580631300820199753, 1.59044695836001970675127076661, 2.82531947651429516884482281212, 4.07732420839011004881163231693, 4.92160580707567218444543087888, 5.58282838826882402879571041131, 5.93621481217910119677661422369, 6.82501020715183982917350035160, 7.83939355453179968057681846324, 8.387982879707162257043738685418
