| L(s) = 1 | − 3-s + 7-s + 9-s + 2.47·11-s + 4.47·13-s + 2·17-s − 6.47·19-s − 21-s + 4·23-s − 27-s − 2·29-s − 10.4·31-s − 2.47·33-s − 10.9·37-s − 4.47·39-s − 2·41-s − 8.94·43-s − 4.94·47-s + 49-s − 2·51-s + 12.4·53-s + 6.47·57-s − 8.94·59-s − 2·61-s + 63-s − 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 0.333·9-s + 0.745·11-s + 1.24·13-s + 0.485·17-s − 1.48·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.371·29-s − 1.88·31-s − 0.430·33-s − 1.79·37-s − 0.716·39-s − 0.312·41-s − 1.36·43-s − 0.721·47-s + 0.142·49-s − 0.280·51-s + 1.71·53-s + 0.857·57-s − 1.16·59-s − 0.256·61-s + 0.125·63-s − 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 - 0.944T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
| 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
−7.23792404474001142606173112574, −6.78047044107963945288656560272, −6.04040870822353242436992552486, −5.44915142013402134740223419999, −4.68499066697667211462748501685, −3.84964467033130007346155416590, −3.33538082495390695378895243912, −1.89246465043237086329901717459, −1.35408281626571590358999963959, 0,
1.35408281626571590358999963959, 1.89246465043237086329901717459, 3.33538082495390695378895243912, 3.84964467033130007346155416590, 4.68499066697667211462748501685, 5.44915142013402134740223419999, 6.04040870822353242436992552486, 6.78047044107963945288656560272, 7.23792404474001142606173112574
