| L(s) = 1 | − 0.262·2-s + 1.16·3-s − 1.93·4-s − 0.0425·5-s − 0.304·6-s + 1.03·8-s − 1.64·9-s + 0.0111·10-s − 6.40·11-s − 2.24·12-s + 2.71·13-s − 0.0494·15-s + 3.59·16-s − 5.19·17-s + 0.432·18-s − 4.81·19-s + 0.0821·20-s + 1.67·22-s − 3.40·23-s + 1.19·24-s − 4.99·25-s − 0.713·26-s − 5.40·27-s + 10.4·29-s + 0.0129·30-s + 8.63·31-s − 3.00·32-s + ⋯ |
| L(s) = 1 | − 0.185·2-s + 0.670·3-s − 0.965·4-s − 0.0190·5-s − 0.124·6-s + 0.364·8-s − 0.549·9-s + 0.00352·10-s − 1.93·11-s − 0.647·12-s + 0.753·13-s − 0.0127·15-s + 0.897·16-s − 1.26·17-s + 0.101·18-s − 1.10·19-s + 0.0183·20-s + 0.358·22-s − 0.710·23-s + 0.244·24-s − 0.999·25-s − 0.139·26-s − 1.03·27-s + 1.93·29-s + 0.00236·30-s + 1.55·31-s − 0.531·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6713 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6713 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7484457988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7484457988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 137 | \( 1 + T \) |
| good | 2 | \( 1 + 0.262T + 2T^{2} \) |
| 3 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 + 0.0425T + 5T^{2} \) |
| 11 | \( 1 + 6.40T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 4.81T + 19T^{2} \) |
| 23 | \( 1 + 3.40T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 8.63T + 31T^{2} \) |
| 37 | \( 1 + 7.34T + 37T^{2} \) |
| 41 | \( 1 - 0.462T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 59 | \( 1 + 6.88T + 59T^{2} \) |
| 61 | \( 1 + 6.30T + 61T^{2} \) |
| 67 | \( 1 + 8.59T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 0.145T + 89T^{2} \) |
| 97 | \( 1 + 8.03T + 97T^{2} \) |
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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
−8.154035904619466373809582602025, −7.71729726478653592266872939228, −6.48284398559786685359917194567, −5.89292310274348347468676935352, −4.95327545797767166923645587938, −4.45568069935176525248998637977, −3.55076308404470510427269423947, −2.70989772439937953122395389498, −2.00811817959369846974978416574, −0.41960420601804905416030110517,
0.41960420601804905416030110517, 2.00811817959369846974978416574, 2.70989772439937953122395389498, 3.55076308404470510427269423947, 4.45568069935176525248998637977, 4.95327545797767166923645587938, 5.89292310274348347468676935352, 6.48284398559786685359917194567, 7.71729726478653592266872939228, 8.154035904619466373809582602025
