| L(s) = 1 | − 0.954·3-s − 1.32·5-s − 2.08·9-s + 2.46·11-s − 1.37·13-s + 1.26·15-s − 2.80·17-s − 19-s + 3.67·23-s − 3.23·25-s + 4.85·27-s − 0.149·29-s + 9.72·31-s − 2.35·33-s − 1.39·37-s + 1.31·39-s − 4.85·41-s + 9.59·43-s + 2.77·45-s − 4.83·47-s + 2.67·51-s + 13.8·53-s − 3.26·55-s + 0.954·57-s − 5.41·59-s − 1.41·61-s + 1.83·65-s + ⋯ |
| L(s) = 1 | − 0.551·3-s − 0.593·5-s − 0.696·9-s + 0.742·11-s − 0.382·13-s + 0.327·15-s − 0.680·17-s − 0.229·19-s + 0.767·23-s − 0.647·25-s + 0.934·27-s − 0.0276·29-s + 1.74·31-s − 0.409·33-s − 0.229·37-s + 0.210·39-s − 0.758·41-s + 1.46·43-s + 0.413·45-s − 0.705·47-s + 0.375·51-s + 1.90·53-s − 0.440·55-s + 0.126·57-s − 0.704·59-s − 0.180·61-s + 0.227·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.954T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 + 0.149T + 29T^{2} \) |
| 31 | \( 1 - 9.72T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 + 4.85T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 0.401T + 73T^{2} \) |
| 79 | \( 1 - 0.228T + 79T^{2} \) |
| 83 | \( 1 + 6.76T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 7.94T + 97T^{2} \) |
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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
−7.47688804377199155898782529314, −6.77062188519355360985566933746, −6.21984388578737166075465488173, −5.46771262559190725017174213783, −4.65540512039145108928933189443, −4.08725730790260868230500188352, −3.13487224920582568963675364822, −2.33197877843186431723000044163, −1.05152595352951820533986564455, 0,
1.05152595352951820533986564455, 2.33197877843186431723000044163, 3.13487224920582568963675364822, 4.08725730790260868230500188352, 4.65540512039145108928933189443, 5.46771262559190725017174213783, 6.21984388578737166075465488173, 6.77062188519355360985566933746, 7.47688804377199155898782529314
