| L(s) = 1 | − 1.96·2-s + 3-s + 1.87·4-s + 3.39·5-s − 1.96·6-s + 0.824·7-s + 0.251·8-s + 9-s − 6.68·10-s − 11-s + 1.87·12-s − 0.512·13-s − 1.62·14-s + 3.39·15-s − 4.23·16-s − 3.00·17-s − 1.96·18-s + 2.20·19-s + 6.35·20-s + 0.824·21-s + 1.96·22-s − 2.99·23-s + 0.251·24-s + 6.53·25-s + 1.00·26-s + 27-s + 1.54·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.577·3-s + 0.936·4-s + 1.51·5-s − 0.803·6-s + 0.311·7-s + 0.0888·8-s + 0.333·9-s − 2.11·10-s − 0.301·11-s + 0.540·12-s − 0.142·13-s − 0.433·14-s + 0.876·15-s − 1.05·16-s − 0.729·17-s − 0.463·18-s + 0.505·19-s + 1.42·20-s + 0.180·21-s + 0.419·22-s − 0.624·23-s + 0.0512·24-s + 1.30·25-s + 0.197·26-s + 0.192·27-s + 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.471530629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471530629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 - 0.824T + 7T^{2} \) |
| 13 | \( 1 + 0.512T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 + 2.99T + 23T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 + 0.419T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 67 | \( 1 + 3.08T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 + 1.99T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.140821026476442420263744713366, −8.574569568784120319904114338661, −7.86049402980721262905835094299, −7.01419764931756800237455017277, −6.23589602441519253380897996313, −5.24636409596157057312514343083, −4.27204565895236944892918467065, −2.64149817914058365648646062587, −2.07152543155714729147844110425, −1.01996438338608687172493985770,
1.01996438338608687172493985770, 2.07152543155714729147844110425, 2.64149817914058365648646062587, 4.27204565895236944892918467065, 5.24636409596157057312514343083, 6.23589602441519253380897996313, 7.01419764931756800237455017277, 7.86049402980721262905835094299, 8.574569568784120319904114338661, 9.140821026476442420263744713366
