L(s) = 1 | + 1.46·2-s − 1.51·3-s + 0.138·4-s − 5-s − 2.20·6-s + 4.07·7-s − 2.72·8-s − 0.718·9-s − 1.46·10-s − 0.621·11-s − 0.208·12-s + 5.09·13-s + 5.95·14-s + 1.51·15-s − 4.25·16-s − 0.266·17-s − 1.05·18-s − 0.138·20-s − 6.14·21-s − 0.908·22-s − 5.84·23-s + 4.11·24-s + 25-s + 7.44·26-s + 5.61·27-s + 0.563·28-s − 4.07·29-s + ⋯ |
L(s) = 1 | + 1.03·2-s − 0.872·3-s + 0.0691·4-s − 0.447·5-s − 0.901·6-s + 1.53·7-s − 0.962·8-s − 0.239·9-s − 0.462·10-s − 0.187·11-s − 0.0603·12-s + 1.41·13-s + 1.59·14-s + 0.389·15-s − 1.06·16-s − 0.0646·17-s − 0.247·18-s − 0.0309·20-s − 1.34·21-s − 0.193·22-s − 1.21·23-s + 0.839·24-s + 0.200·25-s + 1.46·26-s + 1.08·27-s + 0.106·28-s − 0.757·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.949279869\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.949279869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 7 | \( 1 - 4.07T + 7T^{2} \) |
| 11 | \( 1 + 0.621T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 0.266T + 17T^{2} \) |
| 23 | \( 1 + 5.84T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 - 4.48T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 59 | \( 1 + 0.890T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 1.64T + 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 79 | \( 1 + 5.14T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 0.000747T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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
−9.103133210556520497636768980348, −8.335647491255021348953351132534, −7.80021279753426929653795285228, −6.44924788672026532460191486931, −5.83381837801622388050469992896, −5.19726550866396637525091467196, −4.38062898876103329621573459665, −3.79513057237344794271858835850, −2.43384355747603962787730236606, −0.875883791869178934438611308451,
0.875883791869178934438611308451, 2.43384355747603962787730236606, 3.79513057237344794271858835850, 4.38062898876103329621573459665, 5.19726550866396637525091467196, 5.83381837801622388050469992896, 6.44924788672026532460191486931, 7.80021279753426929653795285228, 8.335647491255021348953351132534, 9.103133210556520497636768980348