| L(s) = 1 | + 2-s − 4-s − 3·8-s − 2·11-s − 2·13-s − 16-s − 4·17-s − 2·22-s − 23-s − 2·26-s − 29-s − 2·31-s + 5·32-s − 4·34-s − 2·37-s − 6·41-s + 4·43-s + 2·44-s − 46-s − 7·49-s + 2·52-s − 10·53-s − 58-s + 10·59-s − 2·61-s − 2·62-s + 7·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 0.603·11-s − 0.554·13-s − 1/4·16-s − 0.970·17-s − 0.426·22-s − 0.208·23-s − 0.392·26-s − 0.185·29-s − 0.359·31-s + 0.883·32-s − 0.685·34-s − 0.328·37-s − 0.937·41-s + 0.609·43-s + 0.301·44-s − 0.147·46-s − 49-s + 0.277·52-s − 1.37·53-s − 0.131·58-s + 1.30·59-s − 0.256·61-s − 0.254·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.64766111188247, −13.34516365313308, −13.00286787182391, −12.43827693656069, −12.10134337479455, −11.53724125510284, −10.98298883613349, −10.56692145497327, −9.877022014011717, −9.515536787404065, −9.080152538270336, −8.361732097014120, −8.180177378768952, −7.438030731360577, −6.822646218485482, −6.460581125901733, −5.648068222891640, −5.424156575943270, −4.741723294974553, −4.417933279906166, −3.806598823055156, −3.190772690828770, −2.663332358011974, −2.055259411562859, −1.240139660554179, 0, 0,
1.240139660554179, 2.055259411562859, 2.663332358011974, 3.190772690828770, 3.806598823055156, 4.417933279906166, 4.741723294974553, 5.424156575943270, 5.648068222891640, 6.460581125901733, 6.822646218485482, 7.438030731360577, 8.180177378768952, 8.361732097014120, 9.080152538270336, 9.515536787404065, 9.877022014011717, 10.56692145497327, 10.98298883613349, 11.53724125510284, 12.10134337479455, 12.43827693656069, 13.00286787182391, 13.34516365313308, 13.64766111188247
