| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s − 2·11-s + 4·13-s − 14-s + 16-s − 2·19-s − 2·20-s − 2·22-s − 4·23-s − 25-s + 4·26-s − 28-s + 32-s + 2·35-s + 8·37-s − 2·38-s − 2·40-s − 2·41-s − 4·43-s − 2·44-s − 4·46-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s − 0.603·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 0.447·20-s − 0.426·22-s − 0.834·23-s − 1/5·25-s + 0.784·26-s − 0.188·28-s + 0.176·32-s + 0.338·35-s + 1.31·37-s − 0.324·38-s − 0.316·40-s − 0.312·41-s − 0.609·43-s − 0.301·44-s − 0.589·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.31369377076791, −14.67987331208453, −14.08079950869310, −13.45958536921614, −13.13815758771965, −12.63351846823154, −11.99298447875741, −11.53081811524087, −11.14747295623877, −10.46073539049924, −10.04564014745370, −9.306632900303704, −8.505926403989198, −8.123722453589777, −7.641960764969832, −6.882261980960588, −6.413295276663484, −5.759842501166288, −5.274775146531773, −4.356429807680086, −3.975549536587438, −3.469639087559522, −2.707413786758154, −2.017308339495617, −0.9841195718901139, 0,
0.9841195718901139, 2.017308339495617, 2.707413786758154, 3.469639087559522, 3.975549536587438, 4.356429807680086, 5.274775146531773, 5.759842501166288, 6.413295276663484, 6.882261980960588, 7.641960764969832, 8.123722453589777, 8.505926403989198, 9.306632900303704, 10.04564014745370, 10.46073539049924, 11.14747295623877, 11.53081811524087, 11.99298447875741, 12.63351846823154, 13.13815758771965, 13.45958536921614, 14.08079950869310, 14.67987331208453, 15.31369377076791
