| L(s) = 1 | − 1.17·5-s − 3.14·7-s − 1.67·11-s + 6.63·13-s − 5.16·17-s + 4.72·19-s − 8.82·23-s − 3.63·25-s + 1.80·29-s + 1.65·31-s + 3.68·35-s − 1.99·37-s + 41-s − 1.46·43-s − 8.53·47-s + 2.89·49-s + 9.35·53-s + 1.96·55-s + 8.82·59-s + 12.6·61-s − 7.75·65-s − 9.67·67-s − 0.776·71-s + 8.33·73-s + 5.27·77-s − 0.915·79-s − 10.0·83-s + ⋯ |
| L(s) = 1 | − 0.523·5-s − 1.18·7-s − 0.505·11-s + 1.83·13-s − 1.25·17-s + 1.08·19-s − 1.83·23-s − 0.726·25-s + 0.336·29-s + 0.298·31-s + 0.622·35-s − 0.327·37-s + 0.156·41-s − 0.224·43-s − 1.24·47-s + 0.413·49-s + 1.28·53-s + 0.264·55-s + 1.14·59-s + 1.61·61-s − 0.962·65-s − 1.18·67-s − 0.0921·71-s + 0.975·73-s + 0.600·77-s − 0.103·79-s − 1.10·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082444208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082444208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 - 1.80T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 + 1.99T + 37T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 - 9.35T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.67T + 67T^{2} \) |
| 71 | \( 1 + 0.776T + 71T^{2} \) |
| 73 | \( 1 - 8.33T + 73T^{2} \) |
| 79 | \( 1 + 0.915T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
| 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
−8.279995528763205118939981287379, −7.35816258330228716122818401640, −6.57383770527711878398825648470, −6.08914550637507026063216802990, −5.36153156217550509457871893471, −4.14658366344712037910894585903, −3.73865859262591701421672899731, −2.94808806109812312328813059838, −1.87935567384746022545123085635, −0.53348722595245622583465642529,
0.53348722595245622583465642529, 1.87935567384746022545123085635, 2.94808806109812312328813059838, 3.73865859262591701421672899731, 4.14658366344712037910894585903, 5.36153156217550509457871893471, 6.08914550637507026063216802990, 6.57383770527711878398825648470, 7.35816258330228716122818401640, 8.279995528763205118939981287379
