| L(s) = 1 | + 2.45·2-s + 2.33·3-s + 4.00·4-s − 2.57·5-s + 5.72·6-s + 2.10·7-s + 4.92·8-s + 2.45·9-s − 6.30·10-s − 0.0953·11-s + 9.36·12-s − 0.236·13-s + 5.15·14-s − 6.00·15-s + 4.05·16-s + 17-s + 6.02·18-s − 1.24·19-s − 10.3·20-s + 4.90·21-s − 0.233·22-s − 1.18·23-s + 11.5·24-s + 1.60·25-s − 0.579·26-s − 1.26·27-s + 8.42·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 1.34·3-s + 2.00·4-s − 1.14·5-s + 2.33·6-s + 0.794·7-s + 1.74·8-s + 0.819·9-s − 1.99·10-s − 0.0287·11-s + 2.70·12-s − 0.0655·13-s + 1.37·14-s − 1.55·15-s + 1.01·16-s + 0.242·17-s + 1.42·18-s − 0.285·19-s − 2.30·20-s + 1.07·21-s − 0.0498·22-s − 0.246·23-s + 2.34·24-s + 0.321·25-s − 0.113·26-s − 0.243·27-s + 1.59·28-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
| \(L(1)\) |
\(\approx\) |
\(5.754541747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.754541747\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.45T + 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 + 0.0953T + 11T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 1.18T + 23T^{2} \) |
| 29 | \( 1 - 6.97T + 29T^{2} \) |
| 31 | \( 1 - 0.0214T + 31T^{2} \) |
| 37 | \( 1 + 7.55T + 37T^{2} \) |
| 41 | \( 1 + 0.569T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 - 2.00T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 + 1.16T + 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 3.67T + 79T^{2} \) |
| 83 | \( 1 - 5.62T + 83T^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 - 7.71T + 97T^{2} \) |
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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
−10.15893707442832757282000867752, −8.789286576343847280554011427663, −8.128400608142980057719920131219, −7.47236231319859461507527229331, −6.56978591978603105066683979497, −5.27508724524930656612828953140, −4.44329644863660920538077384537, −3.71851233423377843328578910750, −3.00477865259323120075696200683, −1.93905378778895301173935821180,
1.93905378778895301173935821180, 3.00477865259323120075696200683, 3.71851233423377843328578910750, 4.44329644863660920538077384537, 5.27508724524930656612828953140, 6.56978591978603105066683979497, 7.47236231319859461507527229331, 8.128400608142980057719920131219, 8.789286576343847280554011427663, 10.15893707442832757282000867752
