| L(s) = 1 | − 0.652·2-s − 1.46·3-s − 1.57·4-s − 3.40·5-s + 0.956·6-s + 0.171·7-s + 2.33·8-s − 0.854·9-s + 2.22·10-s + 2.30·12-s + 2.26·13-s − 0.111·14-s + 4.99·15-s + 1.62·16-s + 3.77·17-s + 0.557·18-s + 4.37·19-s + 5.36·20-s − 0.250·21-s + 4.59·23-s − 3.41·24-s + 6.61·25-s − 1.47·26-s + 5.64·27-s − 0.269·28-s + 2.50·29-s − 3.25·30-s + ⋯ |
| L(s) = 1 | − 0.461·2-s − 0.845·3-s − 0.786·4-s − 1.52·5-s + 0.390·6-s + 0.0647·7-s + 0.824·8-s − 0.284·9-s + 0.703·10-s + 0.665·12-s + 0.627·13-s − 0.0298·14-s + 1.28·15-s + 0.406·16-s + 0.915·17-s + 0.131·18-s + 1.00·19-s + 1.19·20-s − 0.0547·21-s + 0.958·23-s − 0.697·24-s + 1.32·25-s − 0.289·26-s + 1.08·27-s − 0.0509·28-s + 0.465·29-s − 0.594·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6928423385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6928423385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.652T + 2T^{2} \) |
| 3 | \( 1 + 1.46T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 - 0.171T + 7T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 - 2.50T + 29T^{2} \) |
| 31 | \( 1 - 9.80T + 31T^{2} \) |
| 37 | \( 1 + 3.50T + 37T^{2} \) |
| 41 | \( 1 + 2.72T + 41T^{2} \) |
| 43 | \( 1 - 5.88T + 43T^{2} \) |
| 47 | \( 1 - 2.76T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 7.41T + 59T^{2} \) |
| 67 | \( 1 - 1.45T + 67T^{2} \) |
| 71 | \( 1 - 1.90T + 71T^{2} \) |
| 73 | \( 1 + 8.05T + 73T^{2} \) |
| 79 | \( 1 + 8.90T + 79T^{2} \) |
| 83 | \( 1 - 0.636T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 3.96T + 97T^{2} \) |
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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
−7.85278416060645175555211485870, −7.46970598019285152832706759742, −6.58395454394228089762465822869, −5.69670768601369424888503375681, −4.99604102494193703543429599772, −4.47019250833285726200066515506, −3.58901852937509727717641986893, −2.99620300413321868517427327679, −1.09793546820857861357111164079, −0.61273102887837577162193270026,
0.61273102887837577162193270026, 1.09793546820857861357111164079, 2.99620300413321868517427327679, 3.58901852937509727717641986893, 4.47019250833285726200066515506, 4.99604102494193703543429599772, 5.69670768601369424888503375681, 6.58395454394228089762465822869, 7.46970598019285152832706759742, 7.85278416060645175555211485870
