L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 1.79·7-s + 8-s + 9-s + 10-s + 4.95·11-s + 12-s + 6.25·13-s − 1.79·14-s + 15-s + 16-s + 18-s − 0.646·19-s + 20-s − 1.79·21-s + 4.95·22-s + 6.06·23-s + 24-s + 25-s + 6.25·26-s + 27-s − 1.79·28-s − 2.33·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.678·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.49·11-s + 0.288·12-s + 1.73·13-s − 0.479·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s − 0.148·19-s + 0.223·20-s − 0.391·21-s + 1.05·22-s + 1.26·23-s + 0.204·24-s + 0.200·25-s + 1.22·26-s + 0.192·27-s − 0.339·28-s − 0.434·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.561484810\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.561484810\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 - 6.25T + 13T^{2} \) |
| 19 | \( 1 + 0.646T + 19T^{2} \) |
| 23 | \( 1 - 6.06T + 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 - 9.41T + 31T^{2} \) |
| 37 | \( 1 - 0.517T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 - 5.95T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 + 9.64T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 2.73T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 5.56T + 73T^{2} \) |
| 79 | \( 1 - 6.63T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 8.67T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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
−7.74457235291397621518300908704, −6.68815132886921685303762084273, −6.43889013209082112780273989249, −5.93772353198347357458621807040, −4.80617712912434407879252071629, −4.18781778526946353457846533108, −3.30102954792571987550359538306, −3.06820852168234939867478973270, −1.72917232740062032021786346736, −1.13005258858345580748457904284,
1.13005258858345580748457904284, 1.72917232740062032021786346736, 3.06820852168234939867478973270, 3.30102954792571987550359538306, 4.18781778526946353457846533108, 4.80617712912434407879252071629, 5.93772353198347357458621807040, 6.43889013209082112780273989249, 6.68815132886921685303762084273, 7.74457235291397621518300908704