| L(s) = 1 | − 3-s − 3.23·5-s − 4.47·7-s + 9-s + 0.763·11-s − 4.47·13-s + 3.23·15-s − 4·17-s + 7.70·19-s + 4.47·21-s − 23-s + 5.47·25-s − 27-s + 4.47·29-s − 6.47·31-s − 0.763·33-s + 14.4·35-s + 6.76·37-s + 4.47·39-s − 2·41-s + 9.23·43-s − 3.23·45-s − 4·47-s + 13.0·49-s + 4·51-s + 0.763·53-s − 2.47·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.44·5-s − 1.69·7-s + 0.333·9-s + 0.230·11-s − 1.24·13-s + 0.835·15-s − 0.970·17-s + 1.76·19-s + 0.975·21-s − 0.208·23-s + 1.09·25-s − 0.192·27-s + 0.830·29-s − 1.16·31-s − 0.132·33-s + 2.44·35-s + 1.11·37-s + 0.716·39-s − 0.312·41-s + 1.40·43-s − 0.482·45-s − 0.583·47-s + 1.85·49-s + 0.560·51-s + 0.104·53-s − 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4741525387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4741525387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9.23T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 0.763T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
| 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
−9.670934882790455958469247534489, −9.339704890922400506819677319505, −8.005609255431561124417733919312, −7.19844251415526251163490800983, −6.71734875753568601192864451470, −5.60947510886304441728161888563, −4.53012720243086103833726658269, −3.66872722980714104157605470220, −2.76638950395827294109009932609, −0.51091552605379783670922286430,
0.51091552605379783670922286430, 2.76638950395827294109009932609, 3.66872722980714104157605470220, 4.53012720243086103833726658269, 5.60947510886304441728161888563, 6.71734875753568601192864451470, 7.19844251415526251163490800983, 8.005609255431561124417733919312, 9.339704890922400506819677319505, 9.670934882790455958469247534489
