| L(s) = 1 | + 2·3-s − 7-s + 9-s + 4·11-s + 4·13-s − 2·17-s + 6·19-s − 2·21-s + 8·23-s − 5·25-s − 4·27-s − 2·29-s − 4·31-s + 8·33-s − 10·37-s + 8·39-s − 10·41-s − 4·43-s + 4·47-s + 49-s − 4·51-s + 2·53-s + 12·57-s − 10·59-s + 8·61-s − 63-s + 8·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.485·17-s + 1.37·19-s − 0.436·21-s + 1.66·23-s − 25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s + 1.39·33-s − 1.64·37-s + 1.28·39-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.560·51-s + 0.274·53-s + 1.58·57-s − 1.30·59-s + 1.02·61-s − 0.125·63-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.054625772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.054625772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−11.18956094201102115432808784878, −9.948797276077021218174011602525, −9.002519157723820939245295475572, −8.733630903561318471582211222656, −7.46301279752361470901499498374, −6.61600017498811391917873139111, −5.38712706105755021847422947230, −3.78995178946695161507197151620, −3.21551153951879944953165713886, −1.60688453692534518598007565808,
1.60688453692534518598007565808, 3.21551153951879944953165713886, 3.78995178946695161507197151620, 5.38712706105755021847422947230, 6.61600017498811391917873139111, 7.46301279752361470901499498374, 8.733630903561318471582211222656, 9.002519157723820939245295475572, 9.948797276077021218174011602525, 11.18956094201102115432808784878
