| L(s) = 1 | + 1.15·2-s + 2.05·3-s − 0.677·4-s + 2.36·6-s − 0.375·7-s − 3.07·8-s + 1.22·9-s − 0.0553·11-s − 1.39·12-s − 0.388·13-s − 0.431·14-s − 2.18·16-s + 1.40·18-s + 1.76·19-s − 0.771·21-s − 0.0636·22-s + 1.02·23-s − 6.32·24-s − 0.446·26-s − 3.64·27-s + 0.254·28-s + 8.45·29-s − 6.05·31-s + 3.64·32-s − 0.113·33-s − 0.829·36-s − 9.49·37-s + ⋯ |
| L(s) = 1 | + 0.813·2-s + 1.18·3-s − 0.338·4-s + 0.965·6-s − 0.141·7-s − 1.08·8-s + 0.408·9-s − 0.0166·11-s − 0.401·12-s − 0.107·13-s − 0.115·14-s − 0.546·16-s + 0.332·18-s + 0.405·19-s − 0.168·21-s − 0.0135·22-s + 0.214·23-s − 1.29·24-s − 0.0876·26-s − 0.702·27-s + 0.0480·28-s + 1.56·29-s − 1.08·31-s + 0.643·32-s − 0.0198·33-s − 0.138·36-s − 1.56·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 7 | \( 1 + 0.375T + 7T^{2} \) |
| 11 | \( 1 + 0.0553T + 11T^{2} \) |
| 13 | \( 1 + 0.388T + 13T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 - 8.45T + 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 + 9.49T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 5.81T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 1.12T + 71T^{2} \) |
| 73 | \( 1 + 2.19T + 73T^{2} \) |
| 79 | \( 1 - 3.94T + 79T^{2} \) |
| 83 | \( 1 - 9.44T + 83T^{2} \) |
| 89 | \( 1 - 2.61T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−7.63884075259623633446756059232, −6.84229509382842268278412562757, −6.12160096022818653281178002360, −5.19225434719845765737270667836, −4.75390344054882532421949774437, −3.69319483653592712303226492912, −3.32716756996508499802703298758, −2.63074677926050812618688573133, −1.58470737411709746085684425796, 0,
1.58470737411709746085684425796, 2.63074677926050812618688573133, 3.32716756996508499802703298758, 3.69319483653592712303226492912, 4.75390344054882532421949774437, 5.19225434719845765737270667836, 6.12160096022818653281178002360, 6.84229509382842268278412562757, 7.63884075259623633446756059232
