| L(s) = 1 | + 2.02·2-s + 3-s + 2.11·4-s + 2.02·6-s − 0.505·7-s + 0.227·8-s + 9-s + 0.687·11-s + 2.11·12-s + 5.78·13-s − 1.02·14-s − 3.76·16-s + 4.74·17-s + 2.02·18-s + 4.23·19-s − 0.505·21-s + 1.39·22-s − 8.36·23-s + 0.227·24-s + 11.7·26-s + 27-s − 1.06·28-s + 4.88·29-s + 2.68·31-s − 8.08·32-s + 0.687·33-s + 9.61·34-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 0.577·3-s + 1.05·4-s + 0.827·6-s − 0.191·7-s + 0.0806·8-s + 0.333·9-s + 0.207·11-s + 0.609·12-s + 1.60·13-s − 0.274·14-s − 0.940·16-s + 1.15·17-s + 0.477·18-s + 0.972·19-s − 0.110·21-s + 0.297·22-s − 1.74·23-s + 0.0465·24-s + 2.30·26-s + 0.192·27-s − 0.201·28-s + 0.907·29-s + 0.482·31-s − 1.42·32-s + 0.119·33-s + 1.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.837284446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.837284446\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 7 | \( 1 + 0.505T + 7T^{2} \) |
| 11 | \( 1 - 0.687T + 11T^{2} \) |
| 13 | \( 1 - 5.78T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 + 8.36T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 - 2.68T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.144T + 41T^{2} \) |
| 43 | \( 1 + 5.94T + 43T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 6.96T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 - 1.31T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 1.17T + 97T^{2} \) |
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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
−9.295381172203025941633211562181, −8.253545193540085540817188554383, −7.70192455159274836446451612754, −6.35004105164461154331123491395, −6.11618610000650750634881043235, −5.06952809216632545323804563902, −4.10061548602113697094931449378, −3.49618856125692233369878862990, −2.75513045792017422822301439607, −1.35194372062325440677981058791,
1.35194372062325440677981058791, 2.75513045792017422822301439607, 3.49618856125692233369878862990, 4.10061548602113697094931449378, 5.06952809216632545323804563902, 6.11618610000650750634881043235, 6.35004105164461154331123491395, 7.70192455159274836446451612754, 8.253545193540085540817188554383, 9.295381172203025941633211562181
