| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s − 7-s − 8-s + 6·9-s − 2·11-s + 3·12-s + 7·13-s + 14-s + 16-s − 4·17-s − 6·18-s − 19-s − 3·21-s + 2·22-s − 3·23-s − 3·24-s − 7·26-s + 9·27-s − 28-s + 9·29-s + 4·31-s − 32-s − 6·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 0.603·11-s + 0.866·12-s + 1.94·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.41·18-s − 0.229·19-s − 0.654·21-s + 0.426·22-s − 0.625·23-s − 0.612·24-s − 1.37·26-s + 1.73·27-s − 0.188·28-s + 1.67·29-s + 0.718·31-s − 0.176·32-s − 1.04·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 617 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.35824736947571, −12.76704311045960, −12.53786776904590, −11.79692721373548, −11.13184888566251, −10.76457364507186, −10.27025336803553, −9.888351193104595, −9.314633543373411, −8.814744493016050, −8.541038015466835, −8.266081653454596, −7.711751919178622, −7.233920154831199, −6.568927229083230, −6.255060134878773, −5.713138512013731, −4.661514103175198, −4.299253703022445, −3.653822206511109, −3.204116819964089, −2.641500165501429, −2.266441350803315, −1.483830411017685, −1.046653160572820, 0,
1.046653160572820, 1.483830411017685, 2.266441350803315, 2.641500165501429, 3.204116819964089, 3.653822206511109, 4.299253703022445, 4.661514103175198, 5.713138512013731, 6.255060134878773, 6.568927229083230, 7.233920154831199, 7.711751919178622, 8.266081653454596, 8.541038015466835, 8.814744493016050, 9.314633543373411, 9.888351193104595, 10.27025336803553, 10.76457364507186, 11.13184888566251, 11.79692721373548, 12.53786776904590, 12.76704311045960, 13.35824736947571
