| L(s) = 1 | − 1.54·2-s − 2.28·3-s + 0.395·4-s + 5-s + 3.52·6-s + 2.48·8-s + 2.19·9-s − 1.54·10-s + 5.50·11-s − 0.902·12-s − 3.50·13-s − 2.28·15-s − 4.63·16-s − 2.26·17-s − 3.40·18-s + 0.659·19-s + 0.395·20-s − 8.52·22-s + 5.48·23-s − 5.66·24-s + 25-s + 5.41·26-s + 1.82·27-s + 1.83·29-s + 3.52·30-s − 31-s + 2.20·32-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 1.31·3-s + 0.197·4-s + 0.447·5-s + 1.44·6-s + 0.877·8-s + 0.732·9-s − 0.489·10-s + 1.66·11-s − 0.260·12-s − 0.970·13-s − 0.588·15-s − 1.15·16-s − 0.548·17-s − 0.802·18-s + 0.151·19-s + 0.0885·20-s − 1.81·22-s + 1.14·23-s − 1.15·24-s + 0.200·25-s + 1.06·26-s + 0.351·27-s + 0.341·29-s + 0.644·30-s − 0.179·31-s + 0.390·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 3 | \( 1 + 2.28T + 3T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 + 2.26T + 17T^{2} \) |
| 19 | \( 1 - 0.659T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 + 6.10T + 41T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 - 0.151T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 6.50T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 6.97T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 9.48T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
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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
−7.37971113771323034926190053735, −6.82187283863631866616147562758, −6.37495366745003931076516600790, −5.48957226973129402651614083854, −4.73642901882112460741461718905, −4.28096811411839343198963720162, −2.95247110434967329733473853239, −1.69810546434328756445825235684, −1.03129352560002431157003841458, 0,
1.03129352560002431157003841458, 1.69810546434328756445825235684, 2.95247110434967329733473853239, 4.28096811411839343198963720162, 4.73642901882112460741461718905, 5.48957226973129402651614083854, 6.37495366745003931076516600790, 6.82187283863631866616147562758, 7.37971113771323034926190053735
