| L(s) = 1 | − 0.714·2-s − 2.62·3-s − 1.48·4-s + 0.882·5-s + 1.87·6-s − 0.838·7-s + 2.49·8-s + 3.89·9-s − 0.630·10-s − 4.52·11-s + 3.91·12-s − 2.33·13-s + 0.599·14-s − 2.31·15-s + 1.19·16-s + 17-s − 2.77·18-s − 3.04·19-s − 1.31·20-s + 2.20·21-s + 3.23·22-s − 7.90·23-s − 6.54·24-s − 4.22·25-s + 1.67·26-s − 2.33·27-s + 1.24·28-s + ⋯ |
| L(s) = 1 | − 0.505·2-s − 1.51·3-s − 0.744·4-s + 0.394·5-s + 0.765·6-s − 0.317·7-s + 0.881·8-s + 1.29·9-s − 0.199·10-s − 1.36·11-s + 1.12·12-s − 0.648·13-s + 0.160·14-s − 0.598·15-s + 0.299·16-s + 0.242·17-s − 0.655·18-s − 0.699·19-s − 0.294·20-s + 0.480·21-s + 0.689·22-s − 1.64·23-s − 1.33·24-s − 0.844·25-s + 0.327·26-s − 0.450·27-s + 0.236·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2892851002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2892851002\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.714T + 2T^{2} \) |
| 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 - 0.882T + 5T^{2} \) |
| 7 | \( 1 + 0.838T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 + 7.90T + 23T^{2} \) |
| 29 | \( 1 + 0.818T + 29T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 - 8.58T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 0.979T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 5.63T + 53T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 5.19T + 71T^{2} \) |
| 73 | \( 1 - 6.63T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 6.61T + 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 + 4.45T + 97T^{2} \) |
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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
−9.928868706558367907072973928967, −9.571843929911018459297820904948, −8.159447749574423325599890205496, −7.60679256047375230565989653729, −6.35507849644508863688595421577, −5.63900984490371141784507942051, −4.96103993135885874915048350494, −4.02621973303133249461557560625, −2.19826368715215087528851670713, −0.46867134377170437727166491262,
0.46867134377170437727166491262, 2.19826368715215087528851670713, 4.02621973303133249461557560625, 4.96103993135885874915048350494, 5.63900984490371141784507942051, 6.35507849644508863688595421577, 7.60679256047375230565989653729, 8.159447749574423325599890205496, 9.571843929911018459297820904948, 9.928868706558367907072973928967
