| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 6·11-s − 13-s + 16-s + 20-s + 6·22-s − 6·23-s + 25-s + 26-s − 4·31-s − 32-s + 10·37-s − 40-s − 8·41-s − 4·43-s − 6·44-s + 6·46-s − 7·49-s − 50-s − 52-s − 10·53-s − 6·55-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.80·11-s − 0.277·13-s + 1/4·16-s + 0.223·20-s + 1.27·22-s − 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.718·31-s − 0.176·32-s + 1.64·37-s − 0.158·40-s − 1.24·41-s − 0.609·43-s − 0.904·44-s + 0.884·46-s − 49-s − 0.141·50-s − 0.138·52-s − 1.37·53-s − 0.809·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.72314903017645, −12.54910059574153, −11.74096895864268, −11.28223145193016, −11.03012175433021, −10.32203057111112, −10.06751105149033, −9.732578322757524, −9.302615300277070, −8.485094073586733, −8.308317820978046, −7.730294161717761, −7.516121746184949, −6.774456995750148, −6.369304773578204, −5.770862244986383, −5.401073497911798, −4.876450296410097, −4.359602336854036, −3.555296080791242, −3.006563878667728, −2.531576495825715, −1.972506553200857, −1.546337785378639, −0.5539854370079836, 0,
0.5539854370079836, 1.546337785378639, 1.972506553200857, 2.531576495825715, 3.006563878667728, 3.555296080791242, 4.359602336854036, 4.876450296410097, 5.401073497911798, 5.770862244986383, 6.369304773578204, 6.774456995750148, 7.516121746184949, 7.730294161717761, 8.308317820978046, 8.485094073586733, 9.302615300277070, 9.732578322757524, 10.06751105149033, 10.32203057111112, 11.03012175433021, 11.28223145193016, 11.74096895864268, 12.54910059574153, 12.72314903017645
