| L(s) = 1 | + 2.78·2-s + 5.78·4-s − 2.33·5-s + 0.430·7-s + 10.5·8-s − 6.51·10-s − 11-s + 13-s + 1.20·14-s + 17.8·16-s + 5.78·17-s − 2.17·19-s − 13.5·20-s − 2.78·22-s + 9.39·23-s + 0.454·25-s + 2.78·26-s + 2.49·28-s + 5.90·29-s − 5.67·31-s + 28.7·32-s + 16.1·34-s − 1.00·35-s − 9.16·37-s − 6.07·38-s − 24.6·40-s + 1.28·41-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 2.89·4-s − 1.04·5-s + 0.162·7-s + 3.72·8-s − 2.06·10-s − 0.301·11-s + 0.277·13-s + 0.321·14-s + 4.46·16-s + 1.40·17-s − 0.499·19-s − 3.01·20-s − 0.594·22-s + 1.95·23-s + 0.0908·25-s + 0.547·26-s + 0.470·28-s + 1.09·29-s − 1.01·31-s + 5.07·32-s + 2.76·34-s − 0.170·35-s − 1.50·37-s − 0.986·38-s − 3.89·40-s + 0.201·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.294962078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.294962078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 5 | \( 1 + 2.33T + 5T^{2} \) |
| 7 | \( 1 - 0.430T + 7T^{2} \) |
| 17 | \( 1 - 5.78T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 - 9.39T + 23T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 6.30T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 - 0.0835T + 67T^{2} \) |
| 71 | \( 1 + 1.97T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 - 6.80T + 79T^{2} \) |
| 83 | \( 1 + 8.93T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 - 4.52T + 97T^{2} \) |
| 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
−10.09722776790011236986994365601, −8.494608651405293908445676028413, −7.65229836540707644575316463190, −7.04173870084363275970696573613, −6.15665077958682999694078280281, −5.12368444779958739494017551888, −4.65868975434446748932721221039, −3.45003926231029265005656498464, −3.15011666730319210494899915241, −1.56534028042961185640815940445,
1.56534028042961185640815940445, 3.15011666730319210494899915241, 3.45003926231029265005656498464, 4.65868975434446748932721221039, 5.12368444779958739494017551888, 6.15665077958682999694078280281, 7.04173870084363275970696573613, 7.65229836540707644575316463190, 8.494608651405293908445676028413, 10.09722776790011236986994365601
