| L(s) = 1 | − 2.49·2-s + 1.20·3-s + 4.23·4-s + 1.95·5-s − 3.00·6-s − 3.56·7-s − 5.58·8-s − 1.55·9-s − 4.87·10-s − 3.71·11-s + 5.09·12-s + 13-s + 8.90·14-s + 2.34·15-s + 5.47·16-s + 4.47·17-s + 3.88·18-s + 4.17·19-s + 8.26·20-s − 4.28·21-s + 9.27·22-s + 2.40·23-s − 6.70·24-s − 1.18·25-s − 2.49·26-s − 5.47·27-s − 15.1·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 0.693·3-s + 2.11·4-s + 0.872·5-s − 1.22·6-s − 1.34·7-s − 1.97·8-s − 0.518·9-s − 1.54·10-s − 1.11·11-s + 1.46·12-s + 0.277·13-s + 2.37·14-s + 0.605·15-s + 1.36·16-s + 1.08·17-s + 0.915·18-s + 0.957·19-s + 1.84·20-s − 0.934·21-s + 1.97·22-s + 0.500·23-s − 1.36·24-s − 0.237·25-s − 0.489·26-s − 1.05·27-s − 2.85·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 - 1.95T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + 6.06T + 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 8.92T + 53T^{2} \) |
| 59 | \( 1 + 4.25T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 1.10T + 67T^{2} \) |
| 71 | \( 1 - 0.746T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 2.59T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 0.496T + 97T^{2} \) |
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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
−9.465345484203136276139850129150, −8.526350418696471919649856569591, −7.84223981345979638391543889215, −7.10380092727362397171066219945, −6.11730736735254297035870975855, −5.44849449214723556677849883867, −3.22961239355034339350927799797, −2.80600582794698915607262246707, −1.59378310913631951746813451292, 0,
1.59378310913631951746813451292, 2.80600582794698915607262246707, 3.22961239355034339350927799797, 5.44849449214723556677849883867, 6.11730736735254297035870975855, 7.10380092727362397171066219945, 7.84223981345979638391543889215, 8.526350418696471919649856569591, 9.465345484203136276139850129150
