| L(s) = 1 | + 1.12·2-s − 1.94·3-s − 0.723·4-s − 2.19·6-s − 2.17·7-s − 3.07·8-s + 0.785·9-s − 4.57·11-s + 1.40·12-s − 2.01·13-s − 2.45·14-s − 2.03·16-s − 1.09·17-s + 0.887·18-s − 7.96·19-s + 4.22·21-s − 5.16·22-s − 6.56·23-s + 5.98·24-s − 2.27·26-s + 4.30·27-s + 1.56·28-s + 0.659·29-s − 7.66·31-s + 3.85·32-s + 8.89·33-s − 1.23·34-s + ⋯ |
| L(s) = 1 | + 0.799·2-s − 1.12·3-s − 0.361·4-s − 0.897·6-s − 0.820·7-s − 1.08·8-s + 0.261·9-s − 1.37·11-s + 0.406·12-s − 0.558·13-s − 0.655·14-s − 0.507·16-s − 0.265·17-s + 0.209·18-s − 1.82·19-s + 0.921·21-s − 1.10·22-s − 1.36·23-s + 1.22·24-s − 0.446·26-s + 0.829·27-s + 0.296·28-s + 0.122·29-s − 1.37·31-s + 0.682·32-s + 1.54·33-s − 0.212·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03257417002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03257417002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 3 | \( 1 + 1.94T + 3T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 + 2.01T + 13T^{2} \) |
| 17 | \( 1 + 1.09T + 17T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 0.659T + 29T^{2} \) |
| 31 | \( 1 + 7.66T + 31T^{2} \) |
| 37 | \( 1 - 5.50T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 - 0.389T + 47T^{2} \) |
| 53 | \( 1 + 3.43T + 53T^{2} \) |
| 59 | \( 1 + 7.35T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 + 3.38T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 + 0.900T + 79T^{2} \) |
| 83 | \( 1 - 0.871T + 83T^{2} \) |
| 89 | \( 1 - 5.48T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−8.468868698688539631400788619137, −7.70044860433552906793018989953, −6.60535948869743259097893295630, −6.06399453705691459557481380959, −5.56106848011250022905779551179, −4.74826620465446041761183002355, −4.18240920695620874817703914849, −3.10581630220116321997908014173, −2.24599440179042283017115187654, −0.090190808268359145276933991089,
0.090190808268359145276933991089, 2.24599440179042283017115187654, 3.10581630220116321997908014173, 4.18240920695620874817703914849, 4.74826620465446041761183002355, 5.56106848011250022905779551179, 6.06399453705691459557481380959, 6.60535948869743259097893295630, 7.70044860433552906793018989953, 8.468868698688539631400788619137
