| L(s) = 1 | − 2.50·2-s + 4.29·4-s − 5-s + 3.13·7-s − 5.77·8-s + 2.50·10-s + 0.248·11-s − 2.67·13-s − 7.86·14-s + 5.88·16-s + 0.902·17-s − 6.19·19-s − 4.29·20-s − 0.622·22-s + 7.69·23-s + 25-s + 6.71·26-s + 13.4·28-s − 2.32·29-s + 7.73·31-s − 3.23·32-s − 2.26·34-s − 3.13·35-s + 3.67·37-s + 15.5·38-s + 5.77·40-s − 9.45·41-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.14·4-s − 0.447·5-s + 1.18·7-s − 2.04·8-s + 0.793·10-s + 0.0748·11-s − 0.742·13-s − 2.10·14-s + 1.47·16-s + 0.218·17-s − 1.42·19-s − 0.961·20-s − 0.132·22-s + 1.60·23-s + 0.200·25-s + 1.31·26-s + 2.54·28-s − 0.432·29-s + 1.38·31-s − 0.571·32-s − 0.388·34-s − 0.529·35-s + 0.604·37-s + 2.52·38-s + 0.912·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7221181308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7221181308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 - 0.248T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 0.902T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 47 | \( 1 - 9.61T + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 - 3.26T + 59T^{2} \) |
| 61 | \( 1 + 3.35T + 61T^{2} \) |
| 67 | \( 1 + 0.613T + 67T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + 1.33T + 73T^{2} \) |
| 79 | \( 1 - 6.65T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 1.34T + 89T^{2} \) |
| 97 | \( 1 - 4.49T + 97T^{2} \) |
| 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
−8.963937550484950104882059662981, −8.488744460002164748552962044772, −7.85591772649711985768523153029, −7.17604999337454316031363966552, −6.48944177826459722230887116124, −5.19915098800863915443291728768, −4.30664922639699012625484929795, −2.79831963251810772040017847192, −1.87264434668958458582353265455, −0.75331960750281953664396671945,
0.75331960750281953664396671945, 1.87264434668958458582353265455, 2.79831963251810772040017847192, 4.30664922639699012625484929795, 5.19915098800863915443291728768, 6.48944177826459722230887116124, 7.17604999337454316031363966552, 7.85591772649711985768523153029, 8.488744460002164748552962044772, 8.963937550484950104882059662981
