| L(s) = 1 | − 3-s + 9-s + 2·11-s − 2·13-s + 2·17-s − 4·19-s − 6·23-s − 27-s + 4·31-s − 2·33-s + 2·37-s + 2·39-s + 6·41-s − 43-s + 2·47-s − 7·49-s − 2·51-s − 14·53-s + 4·57-s − 14·59-s + 14·61-s − 12·67-s + 6·69-s − 6·73-s + 4·79-s + 81-s − 14·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.25·23-s − 0.192·27-s + 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.152·43-s + 0.291·47-s − 49-s − 0.280·51-s − 1.92·53-s + 0.529·57-s − 1.82·59-s + 1.79·61-s − 1.46·67-s + 0.722·69-s − 0.702·73-s + 0.450·79-s + 1/9·81-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8185495508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8185495508\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.03895052945519, −12.43399828993233, −12.03474440939604, −11.81185147806423, −11.02663312618891, −10.89207175796934, −10.06049481107252, −9.919160822360688, −9.353283326839825, −8.811247899114318, −8.221643531377385, −7.734426458471530, −7.379011039363442, −6.590900888497374, −6.272398075222257, −5.922334966257016, −5.252446845443516, −4.583047273255307, −4.355787980698706, −3.700461619833876, −3.042489432932404, −2.404889978331611, −1.721385412159084, −1.195544831116796, −0.2724275871780670,
0.2724275871780670, 1.195544831116796, 1.721385412159084, 2.404889978331611, 3.042489432932404, 3.700461619833876, 4.355787980698706, 4.583047273255307, 5.252446845443516, 5.922334966257016, 6.272398075222257, 6.590900888497374, 7.379011039363442, 7.734426458471530, 8.221643531377385, 8.811247899114318, 9.353283326839825, 9.919160822360688, 10.06049481107252, 10.89207175796934, 11.02663312618891, 11.81185147806423, 12.03474440939604, 12.43399828993233, 13.03895052945519
