| L(s) = 1 | − 1.62·2-s − 1.50·3-s + 0.652·4-s + 1.81·5-s + 2.45·6-s − 1.88·7-s + 2.19·8-s − 0.723·9-s − 2.95·10-s + 3.63·11-s − 0.984·12-s − 4.16·13-s + 3.06·14-s − 2.73·15-s − 4.87·16-s + 17-s + 1.17·18-s − 2.91·19-s + 1.18·20-s + 2.83·21-s − 5.91·22-s + 4.48·23-s − 3.31·24-s − 1.70·25-s + 6.79·26-s + 5.61·27-s − 1.22·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 0.871·3-s + 0.326·4-s + 0.811·5-s + 1.00·6-s − 0.710·7-s + 0.775·8-s − 0.241·9-s − 0.934·10-s + 1.09·11-s − 0.284·12-s − 1.15·13-s + 0.818·14-s − 0.706·15-s − 1.21·16-s + 0.242·17-s + 0.277·18-s − 0.669·19-s + 0.264·20-s + 0.619·21-s − 1.26·22-s + 0.935·23-s − 0.675·24-s − 0.341·25-s + 1.33·26-s + 1.08·27-s − 0.232·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 4.48T + 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 - 0.393T + 31T^{2} \) |
| 37 | \( 1 - 6.72T + 37T^{2} \) |
| 41 | \( 1 + 6.34T + 41T^{2} \) |
| 43 | \( 1 + 1.49T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 + 4.63T + 67T^{2} \) |
| 71 | \( 1 - 4.26T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.722437787591805958291017974321, −8.941806466764188443475128348325, −8.082234700187400917767963849120, −6.81039103438127229369531857042, −6.45591425193977055742555974457, −5.33622604835862734681888062922, −4.43255802142414702573421433603, −2.80116019531676023202934154915, −1.38967294234567652468835695019, 0,
1.38967294234567652468835695019, 2.80116019531676023202934154915, 4.43255802142414702573421433603, 5.33622604835862734681888062922, 6.45591425193977055742555974457, 6.81039103438127229369531857042, 8.082234700187400917767963849120, 8.941806466764188443475128348325, 9.722437787591805958291017974321
