| L(s) = 1 | − 2.75·2-s − 2.26·3-s + 5.57·4-s + 4.03·5-s + 6.24·6-s + 7-s − 9.84·8-s + 2.15·9-s − 11.1·10-s + 5.59·11-s − 12.6·12-s − 6.36·13-s − 2.75·14-s − 9.16·15-s + 15.9·16-s + 3.68·17-s − 5.91·18-s + 0.306·19-s + 22.5·20-s − 2.26·21-s − 15.4·22-s + 4.37·23-s + 22.3·24-s + 11.2·25-s + 17.5·26-s + 1.92·27-s + 5.57·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 1.31·3-s + 2.78·4-s + 1.80·5-s + 2.55·6-s + 0.377·7-s − 3.48·8-s + 0.716·9-s − 3.51·10-s + 1.68·11-s − 3.65·12-s − 1.76·13-s − 0.735·14-s − 2.36·15-s + 3.98·16-s + 0.893·17-s − 1.39·18-s + 0.0702·19-s + 5.03·20-s − 0.495·21-s − 3.28·22-s + 0.912·23-s + 4.56·24-s + 2.25·25-s + 3.43·26-s + 0.371·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8222531430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8222531430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 + 6.36T + 13T^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 - 0.306T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 - 0.873T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 + 8.46T + 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 + 7.33T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 1.49T + 71T^{2} \) |
| 73 | \( 1 - 5.95T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 - 0.537T + 83T^{2} \) |
| 89 | \( 1 - 3.53T + 89T^{2} \) |
| 97 | \( 1 + 9.25T + 97T^{2} \) |
| 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
−8.268346788353110041779256616506, −7.09177823139785530552879906627, −6.80756295231777268491701867913, −6.27472818562187692502531462386, −5.42484484436527523274485047834, −5.02367903539265698841000971422, −3.17764753796730591826974286609, −2.12782089020368586923332132275, −1.50063878978961504970205432299, −0.73437506470412009800950057410,
0.73437506470412009800950057410, 1.50063878978961504970205432299, 2.12782089020368586923332132275, 3.17764753796730591826974286609, 5.02367903539265698841000971422, 5.42484484436527523274485047834, 6.27472818562187692502531462386, 6.80756295231777268491701867913, 7.09177823139785530552879906627, 8.268346788353110041779256616506
