| L(s) = 1 | − 0.751·2-s + 2.33·3-s − 1.43·4-s + 4.26·5-s − 1.75·6-s + 2.58·8-s + 2.43·9-s − 3.20·10-s + 4.09·11-s − 3.34·12-s − 2.18·13-s + 9.94·15-s + 0.933·16-s + 0.590·17-s − 1.82·18-s − 19-s − 6.12·20-s − 3.07·22-s − 4.36·23-s + 6.01·24-s + 13.1·25-s + 1.64·26-s − 1.31·27-s − 7.36·29-s − 7.46·30-s + 4.51·31-s − 5.86·32-s + ⋯ |
| L(s) = 1 | − 0.531·2-s + 1.34·3-s − 0.717·4-s + 1.90·5-s − 0.714·6-s + 0.912·8-s + 0.811·9-s − 1.01·10-s + 1.23·11-s − 0.966·12-s − 0.606·13-s + 2.56·15-s + 0.233·16-s + 0.143·17-s − 0.431·18-s − 0.229·19-s − 1.36·20-s − 0.656·22-s − 0.911·23-s + 1.22·24-s + 2.63·25-s + 0.322·26-s − 0.253·27-s − 1.36·29-s − 1.36·30-s + 0.811·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.279112326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279112326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.751T + 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 11 | \( 1 - 4.09T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 - 0.590T + 17T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 - 4.51T + 31T^{2} \) |
| 37 | \( 1 - 8.95T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 4.40T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.32T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 - 3.79T + 89T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−9.771347311326652050871311684902, −9.299279976632720186461755706697, −8.684865800503177075821312143817, −7.88041095356055227509387985460, −6.73981821717403209949099307269, −5.79424059970107611880871651829, −4.70027749766782318089513678246, −3.56846096220388024907985869952, −2.28848590950889300415816273341, −1.48991594971378200226454616630,
1.48991594971378200226454616630, 2.28848590950889300415816273341, 3.56846096220388024907985869952, 4.70027749766782318089513678246, 5.79424059970107611880871651829, 6.73981821717403209949099307269, 7.88041095356055227509387985460, 8.684865800503177075821312143817, 9.299279976632720186461755706697, 9.771347311326652050871311684902
