| L(s) = 1 | + 1.98·2-s + 3-s + 1.93·4-s + 5-s + 1.98·6-s − 3.28·7-s − 0.129·8-s + 9-s + 1.98·10-s − 1.92·11-s + 1.93·12-s − 4.28·13-s − 6.52·14-s + 15-s − 4.12·16-s + 4.24·17-s + 1.98·18-s + 8.38·19-s + 1.93·20-s − 3.28·21-s − 3.82·22-s − 0.129·24-s + 25-s − 8.49·26-s + 27-s − 6.36·28-s − 4.27·29-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.577·3-s + 0.967·4-s + 0.447·5-s + 0.809·6-s − 1.24·7-s − 0.0458·8-s + 0.333·9-s + 0.627·10-s − 0.580·11-s + 0.558·12-s − 1.18·13-s − 1.74·14-s + 0.258·15-s − 1.03·16-s + 1.02·17-s + 0.467·18-s + 1.92·19-s + 0.432·20-s − 0.717·21-s − 0.814·22-s − 0.0264·24-s + 0.200·25-s − 1.66·26-s + 0.192·27-s − 1.20·28-s − 0.794·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 4.28T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 - 8.38T + 19T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 9.32T + 37T^{2} \) |
| 41 | \( 1 + 9.91T + 41T^{2} \) |
| 43 | \( 1 + 9.01T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 + 6.65T + 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 + 0.103T + 73T^{2} \) |
| 79 | \( 1 - 5.86T + 79T^{2} \) |
| 83 | \( 1 + 4.70T + 83T^{2} \) |
| 89 | \( 1 - 8.91T + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
| 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
−7.39788871838668510199666370437, −6.68943958381269461640019422685, −5.84788146686966196872762546380, −5.36316085705869071353608146745, −4.77667472475482483684328631213, −3.71959728798133324901248902846, −3.09318332279673325704362089864, −2.82229820573538767778295941995, −1.67228004898470511080587636453, 0,
1.67228004898470511080587636453, 2.82229820573538767778295941995, 3.09318332279673325704362089864, 3.71959728798133324901248902846, 4.77667472475482483684328631213, 5.36316085705869071353608146745, 5.84788146686966196872762546380, 6.68943958381269461640019422685, 7.39788871838668510199666370437
