| L(s) = 1 | + 2·5-s − 7-s + 2·11-s + 2·13-s + 6·17-s − 4·19-s + 6·23-s − 25-s − 4·31-s − 2·35-s + 10·37-s + 2·41-s − 4·43-s + 4·47-s + 49-s − 12·53-s + 4·55-s + 12·59-s + 6·61-s + 4·65-s − 4·67-s − 14·71-s − 2·73-s − 2·77-s − 8·79-s − 16·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.603·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s − 0.718·31-s − 0.338·35-s + 1.64·37-s + 0.312·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.64·53-s + 0.539·55-s + 1.56·59-s + 0.768·61-s + 0.496·65-s − 0.488·67-s − 1.66·71-s − 0.234·73-s − 0.227·77-s − 0.900·79-s − 1.75·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.698425554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698425554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−10.84603417852077997548476185489, −9.915934682408437918756969494868, −9.302111474223371601690156394253, −8.348093232759484287064194714284, −7.16759940438630821447644837255, −6.18921559649042192845851641510, −5.50301216544551056360312768488, −4.10502926185056205242132098716, −2.88871942698849130206456913367, −1.38744136941764418419874219848,
1.38744136941764418419874219848, 2.88871942698849130206456913367, 4.10502926185056205242132098716, 5.50301216544551056360312768488, 6.18921559649042192845851641510, 7.16759940438630821447644837255, 8.348093232759484287064194714284, 9.302111474223371601690156394253, 9.915934682408437918756969494868, 10.84603417852077997548476185489
