| L(s) = 1 | + 2-s − 3-s + 4-s − 1.73·5-s − 6-s − 1.26·7-s + 8-s + 9-s − 1.73·10-s − 1.26·11-s − 12-s − 1.26·14-s + 1.73·15-s + 16-s + 5.19·17-s + 18-s − 4.73·19-s − 1.73·20-s + 1.26·21-s − 1.26·22-s − 8.19·23-s − 24-s − 2.00·25-s − 27-s − 1.26·28-s − 3·29-s + 1.73·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.774·5-s − 0.408·6-s − 0.479·7-s + 0.353·8-s + 0.333·9-s − 0.547·10-s − 0.382·11-s − 0.288·12-s − 0.338·14-s + 0.447·15-s + 0.250·16-s + 1.26·17-s + 0.235·18-s − 1.08·19-s − 0.387·20-s + 0.276·21-s − 0.270·22-s − 1.70·23-s − 0.204·24-s − 0.400·25-s − 0.192·27-s − 0.239·28-s − 0.557·29-s + 0.316·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 6.46T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.899936108553723742336988602302, −8.519425445042401951191407361988, −7.68908503613693334215394136148, −6.93111959358357448252329457893, −5.92281481388192298280315888349, −5.30046054136040800899147906820, −4.05742191976506767341598241034, −3.52144566342169011415882278305, −1.99104777947687867489605592496, 0,
1.99104777947687867489605592496, 3.52144566342169011415882278305, 4.05742191976506767341598241034, 5.30046054136040800899147906820, 5.92281481388192298280315888349, 6.93111959358357448252329457893, 7.68908503613693334215394136148, 8.519425445042401951191407361988, 9.899936108553723742336988602302
