| L(s) = 1 | − 1.30·2-s − 1.88·3-s − 0.287·4-s + 2.46·6-s + 1.28·7-s + 2.99·8-s + 0.550·9-s + 0.698·11-s + 0.541·12-s − 5.53·13-s − 1.68·14-s − 3.34·16-s + 3.43·17-s − 0.720·18-s − 3.98·19-s − 2.42·21-s − 0.914·22-s + 6.27·23-s − 5.64·24-s + 7.24·26-s + 4.61·27-s − 0.369·28-s + 6.15·29-s − 0.212·31-s − 1.61·32-s − 1.31·33-s − 4.49·34-s + ⋯ |
| L(s) = 1 | − 0.925·2-s − 1.08·3-s − 0.143·4-s + 1.00·6-s + 0.486·7-s + 1.05·8-s + 0.183·9-s + 0.210·11-s + 0.156·12-s − 1.53·13-s − 0.450·14-s − 0.835·16-s + 0.832·17-s − 0.169·18-s − 0.914·19-s − 0.529·21-s − 0.195·22-s + 1.30·23-s − 1.15·24-s + 1.42·26-s + 0.888·27-s − 0.0698·28-s + 1.14·29-s − 0.0381·31-s − 0.284·32-s − 0.229·33-s − 0.770·34-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 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 + 1.88T + 3T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 - 0.698T + 11T^{2} \) |
| 13 | \( 1 + 5.53T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 23 | \( 1 - 6.27T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 + 0.212T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 + 7.78T + 43T^{2} \) |
| 47 | \( 1 - 3.86T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + 6.00T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 + 6.80T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 - 0.714T + 97T^{2} \) |
| 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
−7.79551109451281616116857541655, −7.00122894964373722652844936373, −6.56103786275218902083376569004, −5.28536340655025904300428686251, −5.07584419706933863171416824840, −4.36789249536902137071058087046, −3.12530584994224424905513951226, −1.95590663826772561839969901111, −0.941584792792833557930637510237, 0,
0.941584792792833557930637510237, 1.95590663826772561839969901111, 3.12530584994224424905513951226, 4.36789249536902137071058087046, 5.07584419706933863171416824840, 5.28536340655025904300428686251, 6.56103786275218902083376569004, 7.00122894964373722652844936373, 7.79551109451281616116857541655
