| L(s) = 1 | − 2·5-s − 3·9-s + 4·11-s − 2·13-s + 6·17-s + 8·19-s − 25-s + 6·29-s + 8·31-s − 2·37-s − 2·41-s + 4·43-s + 6·45-s − 8·47-s + 6·53-s − 8·55-s + 6·61-s + 4·65-s + 4·67-s + 8·71-s − 10·73-s − 16·79-s + 9·81-s + 8·83-s − 12·85-s + 6·89-s − 16·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 0.824·53-s − 1.07·55-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 81-s + 0.878·83-s − 1.30·85-s + 0.635·89-s − 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322192770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322192770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.15969982113999416146742248886, −9.502244133602489113553033388660, −8.459138850523211127936437645267, −7.80607013998257452679138271336, −6.92383462522053566355627597037, −5.84465028770243608573987581113, −4.90615311225424089857668766077, −3.69860481691342078568866024595, −2.92766858317331148516836536443, −0.988078077776060913977316087411,
0.988078077776060913977316087411, 2.92766858317331148516836536443, 3.69860481691342078568866024595, 4.90615311225424089857668766077, 5.84465028770243608573987581113, 6.92383462522053566355627597037, 7.80607013998257452679138271336, 8.459138850523211127936437645267, 9.502244133602489113553033388660, 10.15969982113999416146742248886
