| L(s) = 1 | − 1.74·2-s + 1.04·4-s − 0.892·5-s + 3.89·7-s + 1.66·8-s + 1.55·10-s − 4.47·11-s + 2.21·13-s − 6.79·14-s − 4.99·16-s + 5.25·17-s + 0.817·19-s − 0.931·20-s + 7.80·22-s − 23-s − 4.20·25-s − 3.86·26-s + 4.06·28-s − 29-s − 4.19·31-s + 5.38·32-s − 9.17·34-s − 3.47·35-s + 8.35·37-s − 1.42·38-s − 1.48·40-s + 1.95·41-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.521·4-s − 0.399·5-s + 1.47·7-s + 0.589·8-s + 0.492·10-s − 1.34·11-s + 0.613·13-s − 1.81·14-s − 1.24·16-s + 1.27·17-s + 0.187·19-s − 0.208·20-s + 1.66·22-s − 0.208·23-s − 0.840·25-s − 0.757·26-s + 0.768·28-s − 0.185·29-s − 0.753·31-s + 0.951·32-s − 1.57·34-s − 0.587·35-s + 1.37·37-s − 0.231·38-s − 0.235·40-s + 0.305·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9874394389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9874394389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 5 | \( 1 + 0.892T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 - 5.25T + 17T^{2} \) |
| 19 | \( 1 - 0.817T + 19T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 - 0.517T + 53T^{2} \) |
| 59 | \( 1 - 6.99T + 59T^{2} \) |
| 61 | \( 1 + 0.394T + 61T^{2} \) |
| 67 | \( 1 - 7.01T + 67T^{2} \) |
| 71 | \( 1 - 4.42T + 71T^{2} \) |
| 73 | \( 1 + 0.726T + 73T^{2} \) |
| 79 | \( 1 + 7.50T + 79T^{2} \) |
| 83 | \( 1 + 8.09T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
| 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
−8.004139663884253314929620668182, −7.77524080201193233272497239495, −7.17824925455069225904892951816, −5.86191061532580786891722397382, −5.25479546434276700346160692775, −4.50472862013452949201877073451, −3.65335423035840606924657993767, −2.42426933301993003874237589229, −1.58205255772384485868978855845, −0.66833197979268433995452391988,
0.66833197979268433995452391988, 1.58205255772384485868978855845, 2.42426933301993003874237589229, 3.65335423035840606924657993767, 4.50472862013452949201877073451, 5.25479546434276700346160692775, 5.86191061532580786891722397382, 7.17824925455069225904892951816, 7.77524080201193233272497239495, 8.004139663884253314929620668182
