| L(s) = 1 | + 2·2-s + 3·3-s + 2·4-s + 5-s + 6·6-s − 7-s + 6·9-s + 2·10-s − 11-s + 6·12-s − 2·13-s − 2·14-s + 3·15-s − 4·16-s + 17-s + 12·18-s + 8·19-s + 2·20-s − 3·21-s − 2·22-s + 23-s − 4·25-s − 4·26-s + 9·27-s − 2·28-s + 6·30-s − 5·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 4-s + 0.447·5-s + 2.44·6-s − 0.377·7-s + 2·9-s + 0.632·10-s − 0.301·11-s + 1.73·12-s − 0.554·13-s − 0.534·14-s + 0.774·15-s − 16-s + 0.242·17-s + 2.82·18-s + 1.83·19-s + 0.447·20-s − 0.654·21-s − 0.426·22-s + 0.208·23-s − 4/5·25-s − 0.784·26-s + 1.73·27-s − 0.377·28-s + 1.09·30-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.048704970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.048704970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.446438931441163168534625971809, −9.061383627300795542014436143149, −7.76186528671323390349193545818, −7.33329936418067495191636779077, −6.15662682551987716910344507203, −5.26011452170419405394373489317, −4.33476172488371417519220981259, −3.30923706983684727463664235260, −2.92859443338052510588572354071, −1.85282375092272145420962474837,
1.85282375092272145420962474837, 2.92859443338052510588572354071, 3.30923706983684727463664235260, 4.33476172488371417519220981259, 5.26011452170419405394373489317, 6.15662682551987716910344507203, 7.33329936418067495191636779077, 7.76186528671323390349193545818, 9.061383627300795542014436143149, 9.446438931441163168534625971809
