L(s) = 1 | − 2.80·2-s + 0.485·3-s + 5.86·4-s − 1.36·6-s + 4.89·7-s − 10.8·8-s − 2.76·9-s + 0.293·11-s + 2.84·12-s − 5.26·13-s − 13.7·14-s + 18.6·16-s + 1.23·17-s + 7.75·18-s + 0.414·19-s + 2.37·21-s − 0.822·22-s − 8.19·23-s − 5.26·24-s + 14.7·26-s − 2.79·27-s + 28.7·28-s + 4.60·29-s − 3.52·31-s − 30.7·32-s + 0.142·33-s − 3.45·34-s + ⋯ |
L(s) = 1 | − 1.98·2-s + 0.280·3-s + 2.93·4-s − 0.556·6-s + 1.84·7-s − 3.83·8-s − 0.921·9-s + 0.0884·11-s + 0.822·12-s − 1.45·13-s − 3.66·14-s + 4.67·16-s + 0.298·17-s + 1.82·18-s + 0.0950·19-s + 0.518·21-s − 0.175·22-s − 1.70·23-s − 1.07·24-s + 2.89·26-s − 0.538·27-s + 5.42·28-s + 0.854·29-s − 0.633·31-s − 5.43·32-s + 0.0247·33-s − 0.592·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 3 | \( 1 - 0.485T + 3T^{2} \) |
| 7 | \( 1 - 4.89T + 7T^{2} \) |
| 11 | \( 1 - 0.293T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 0.414T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 + 3.52T + 31T^{2} \) |
| 37 | \( 1 + 2.98T + 37T^{2} \) |
| 41 | \( 1 - 3.04T + 41T^{2} \) |
| 43 | \( 1 + 1.50T + 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 - 8.04T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 8.20T + 71T^{2} \) |
| 73 | \( 1 + 1.83T + 73T^{2} \) |
| 79 | \( 1 - 7.62T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 0.911T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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
−8.064817644044215722989151174330, −7.43768296563644621268906847659, −6.71658242548787029918739755206, −5.68205996573467541573265613837, −5.14863069409627964895532714053, −3.80893037693856543605434208047, −2.48465988686890562329742316982, −2.21017920605599208829316340529, −1.22001902343528743429347017982, 0,
1.22001902343528743429347017982, 2.21017920605599208829316340529, 2.48465988686890562329742316982, 3.80893037693856543605434208047, 5.14863069409627964895532714053, 5.68205996573467541573265613837, 6.71658242548787029918739755206, 7.43768296563644621268906847659, 8.064817644044215722989151174330