| L(s) = 1 | − 2.46·2-s + 2.84·3-s + 4.05·4-s − 0.414·5-s − 7.00·6-s − 0.439·7-s − 5.06·8-s + 5.10·9-s + 1.02·10-s + 4.90·11-s + 11.5·12-s − 0.372·13-s + 1.08·14-s − 1.18·15-s + 4.34·16-s + 17-s − 12.5·18-s − 1.13·19-s − 1.68·20-s − 1.25·21-s − 12.0·22-s + 2.97·23-s − 14.4·24-s − 4.82·25-s + 0.917·26-s + 5.99·27-s − 1.78·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 1.64·3-s + 2.02·4-s − 0.185·5-s − 2.86·6-s − 0.166·7-s − 1.78·8-s + 1.70·9-s + 0.322·10-s + 1.48·11-s + 3.33·12-s − 0.103·13-s + 0.288·14-s − 0.304·15-s + 1.08·16-s + 0.242·17-s − 2.96·18-s − 0.261·19-s − 0.376·20-s − 0.272·21-s − 2.57·22-s + 0.620·23-s − 2.94·24-s − 0.965·25-s + 0.179·26-s + 1.15·27-s − 0.336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343065464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343065464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 + 0.414T + 5T^{2} \) |
| 7 | \( 1 + 0.439T + 7T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 0.227T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 3.23T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 - 7.42T + 47T^{2} \) |
| 53 | \( 1 - 1.45T + 53T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 + 1.80T + 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 - 2.38T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 - 7.33T + 97T^{2} \) |
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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
−9.530365246292580875772215516532, −9.207317560333154021874192215142, −8.401887035033826434395653072924, −7.88018328939903372304665825427, −7.04929451929833695363501610036, −6.30469202216851707746077707116, −4.29111596007868846749709816503, −3.25632568836287053371862640686, −2.23651546709291763649843602633, −1.17969855166627381769326767105,
1.17969855166627381769326767105, 2.23651546709291763649843602633, 3.25632568836287053371862640686, 4.29111596007868846749709816503, 6.30469202216851707746077707116, 7.04929451929833695363501610036, 7.88018328939903372304665825427, 8.401887035033826434395653072924, 9.207317560333154021874192215142, 9.530365246292580875772215516532
