L(s) = 1 | + 2-s + 4-s + 5-s + 4.54·7-s + 8-s + 10-s − 2.54·11-s − 1.05·13-s + 4.54·14-s + 16-s − 4.39·17-s + 7.60·19-s + 20-s − 2.54·22-s − 1.20·23-s + 25-s − 1.05·26-s + 4.54·28-s + 6.10·29-s + 31-s + 32-s − 4.39·34-s + 4.54·35-s + 7.73·37-s + 7.60·38-s + 40-s + 6.10·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.71·7-s + 0.353·8-s + 0.316·10-s − 0.768·11-s − 0.291·13-s + 1.21·14-s + 0.250·16-s − 1.06·17-s + 1.74·19-s + 0.223·20-s − 0.543·22-s − 0.251·23-s + 0.200·25-s − 0.206·26-s + 0.859·28-s + 1.13·29-s + 0.179·31-s + 0.176·32-s − 0.753·34-s + 0.769·35-s + 1.27·37-s + 1.23·38-s + 0.158·40-s + 0.953·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.839811704\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.839811704\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 4.54T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 - 7.60T + 19T^{2} \) |
| 23 | \( 1 + 1.20T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 - 6.10T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 2.68T + 47T^{2} \) |
| 53 | \( 1 + 0.502T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 9.80T + 61T^{2} \) |
| 67 | \( 1 - 7.46T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 73 | \( 1 + 8.70T + 73T^{2} \) |
| 79 | \( 1 + 0.918T + 79T^{2} \) |
| 83 | \( 1 - 2.41T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 9.20T + 97T^{2} \) |
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
−8.692344372131066392565378423263, −7.81530316257676966477863733405, −7.46191443175957172072765419525, −6.34769003131846049369736758001, −5.52757077701372068293301620199, −4.83580789590666717890633900747, −4.41394022223565056661905164620, −3.00214744359262292638309022445, −2.22114055419574764647340607535, −1.21060833244101024550659073358,
1.21060833244101024550659073358, 2.22114055419574764647340607535, 3.00214744359262292638309022445, 4.41394022223565056661905164620, 4.83580789590666717890633900747, 5.52757077701372068293301620199, 6.34769003131846049369736758001, 7.46191443175957172072765419525, 7.81530316257676966477863733405, 8.692344372131066392565378423263