| L(s) = 1 | + 1.00·2-s − 2.57·3-s − 0.981·4-s − 2.22·5-s − 2.59·6-s − 3.29·7-s − 3.00·8-s + 3.61·9-s − 2.24·10-s − 4.50·11-s + 2.52·12-s + 2.38·13-s − 3.32·14-s + 5.73·15-s − 1.07·16-s − 17-s + 3.64·18-s − 2.30·19-s + 2.18·20-s + 8.47·21-s − 4.55·22-s + 0.109·23-s + 7.73·24-s − 0.0314·25-s + 2.40·26-s − 1.57·27-s + 3.23·28-s + ⋯ |
| L(s) = 1 | + 0.713·2-s − 1.48·3-s − 0.490·4-s − 0.996·5-s − 1.05·6-s − 1.24·7-s − 1.06·8-s + 1.20·9-s − 0.711·10-s − 1.35·11-s + 0.728·12-s + 0.660·13-s − 0.888·14-s + 1.48·15-s − 0.268·16-s − 0.242·17-s + 0.859·18-s − 0.528·19-s + 0.489·20-s + 1.84·21-s − 0.970·22-s + 0.0229·23-s + 1.57·24-s − 0.00629·25-s + 0.471·26-s − 0.303·27-s + 0.611·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2425273203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2425273203\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.00T + 2T^{2} \) |
| 3 | \( 1 + 2.57T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 - 0.109T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 0.826T + 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 - 2.52T + 53T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 0.0674T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 0.237T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 5.16T + 97T^{2} \) |
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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
−10.25518060291342126667146907739, −9.243036450051958311954440747425, −8.233225680543310916876854273807, −7.19291945075860369987204224270, −6.19377814640834339048363695585, −5.69947768434816488569494839744, −4.74241411909662612022287085706, −3.95570645253799974825576876527, −2.97993634133098499257426470069, −0.34908738473829809323043856137,
0.34908738473829809323043856137, 2.97993634133098499257426470069, 3.95570645253799974825576876527, 4.74241411909662612022287085706, 5.69947768434816488569494839744, 6.19377814640834339048363695585, 7.19291945075860369987204224270, 8.233225680543310916876854273807, 9.243036450051958311954440747425, 10.25518060291342126667146907739
