| L(s) = 1 | − 2.26·2-s − 3-s + 3.10·4-s + 2.26·6-s − 4.96·7-s − 2.50·8-s + 9-s − 3.21·11-s − 3.10·12-s − 13-s + 11.2·14-s − 0.551·16-s + 0.448·17-s − 2.26·18-s − 7.23·19-s + 4.96·21-s + 7.27·22-s + 6.26·23-s + 2.50·24-s + 2.26·26-s − 27-s − 15.4·28-s − 2.21·29-s + 2.19·31-s + 6.26·32-s + 3.21·33-s − 1.01·34-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.55·4-s + 0.922·6-s − 1.87·7-s − 0.886·8-s + 0.333·9-s − 0.969·11-s − 0.897·12-s − 0.277·13-s + 3.00·14-s − 0.137·16-s + 0.108·17-s − 0.532·18-s − 1.65·19-s + 1.08·21-s + 1.55·22-s + 1.30·23-s + 0.511·24-s + 0.443·26-s − 0.192·27-s − 2.92·28-s − 0.411·29-s + 0.393·31-s + 1.10·32-s + 0.559·33-s − 0.173·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2129449935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2129449935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 7 | \( 1 + 4.96T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 17 | \( 1 - 0.448T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 6.26T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 8.26T + 37T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 + 6.21T + 43T^{2} \) |
| 47 | \( 1 + 1.66T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 2.73T + 67T^{2} \) |
| 71 | \( 1 - 4.11T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.05T + 79T^{2} \) |
| 83 | \( 1 - 5.77T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 8.37T + 97T^{2} \) |
| 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
−10.05368108413244169896812037135, −9.248719866130353739861225313157, −8.568474640207515518483969097745, −7.49712708528509198181571424788, −6.75077499878738586063400317291, −6.18725036055468602674322244935, −4.90118879501670823821850660236, −3.35842319668086697588695853900, −2.22371035960349399474110333887, −0.44050679013520861333993073006,
0.44050679013520861333993073006, 2.22371035960349399474110333887, 3.35842319668086697588695853900, 4.90118879501670823821850660236, 6.18725036055468602674322244935, 6.75077499878738586063400317291, 7.49712708528509198181571424788, 8.568474640207515518483969097745, 9.248719866130353739861225313157, 10.05368108413244169896812037135
