L(s) = 1 | + 1.21·2-s − 0.512·4-s − 5-s − 3.60·7-s − 3.06·8-s − 1.21·10-s + 5.37·11-s − 4.39·14-s − 2.71·16-s + 1.13·17-s − 2.26·19-s + 0.512·20-s + 6.55·22-s + 3.89·23-s + 25-s + 1.84·28-s + 0.0247·29-s − 5.46·31-s + 2.81·32-s + 1.38·34-s + 3.60·35-s + 8.70·37-s − 2.76·38-s + 3.06·40-s + 3.73·41-s − 1.13·43-s − 2.75·44-s + ⋯ |
L(s) = 1 | + 0.862·2-s − 0.256·4-s − 0.447·5-s − 1.36·7-s − 1.08·8-s − 0.385·10-s + 1.61·11-s − 1.17·14-s − 0.678·16-s + 0.274·17-s − 0.520·19-s + 0.114·20-s + 1.39·22-s + 0.811·23-s + 0.200·25-s + 0.348·28-s + 0.00459·29-s − 0.981·31-s + 0.498·32-s + 0.236·34-s + 0.608·35-s + 1.43·37-s − 0.448·38-s + 0.484·40-s + 0.582·41-s − 0.172·43-s − 0.414·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 + 2.26T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 0.0247T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 8.70T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 - 2.58T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 0.171T + 59T^{2} \) |
| 61 | \( 1 - 3.36T + 61T^{2} \) |
| 67 | \( 1 + 6.39T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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
−7.20317648226588894878435850138, −6.73776527565038545381999542512, −6.03896304691145738748897444447, −5.53587662728401401127953027312, −4.33782133692622593527616771527, −4.07590574891540434307633999927, −3.31251905765708186129095668421, −2.69418708577891787481258806412, −1.16615682181981048647370665586, 0,
1.16615682181981048647370665586, 2.69418708577891787481258806412, 3.31251905765708186129095668421, 4.07590574891540434307633999927, 4.33782133692622593527616771527, 5.53587662728401401127953027312, 6.03896304691145738748897444447, 6.73776527565038545381999542512, 7.20317648226588894878435850138