| L(s) = 1 | − 2-s + 0.490·3-s + 4-s − 0.490·6-s − 7-s − 8-s − 2.75·9-s + 3.31·11-s + 0.490·12-s − 2.44·13-s + 14-s + 16-s − 17-s + 2.75·18-s + 2.77·19-s − 0.490·21-s − 3.31·22-s + 2.58·23-s − 0.490·24-s + 2.44·26-s − 2.82·27-s − 28-s + 9.02·29-s − 5.40·31-s − 32-s + 1.62·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.282·3-s + 0.5·4-s − 0.200·6-s − 0.377·7-s − 0.353·8-s − 0.919·9-s + 0.998·11-s + 0.141·12-s − 0.678·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s + 0.650·18-s + 0.637·19-s − 0.106·21-s − 0.706·22-s + 0.538·23-s − 0.100·24-s + 0.479·26-s − 0.543·27-s − 0.188·28-s + 1.67·29-s − 0.970·31-s − 0.176·32-s + 0.282·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.490T + 3T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 19 | \( 1 - 2.77T + 19T^{2} \) |
| 23 | \( 1 - 2.58T + 23T^{2} \) |
| 29 | \( 1 - 9.02T + 29T^{2} \) |
| 31 | \( 1 + 5.40T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 + 0.332T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 - 6.05T + 61T^{2} \) |
| 67 | \( 1 + 1.66T + 67T^{2} \) |
| 71 | \( 1 - 6.40T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 8.49T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.906627606183617297685905366073, −6.81282884808766437212669574833, −6.76328826156422739602668060433, −5.62852778629864333793496030835, −4.98204825486442364240106106447, −3.79980201197211831829922014328, −3.10552209082498516932405023020, −2.32233058808145223154033822683, −1.23550145037630326934403883151, 0,
1.23550145037630326934403883151, 2.32233058808145223154033822683, 3.10552209082498516932405023020, 3.79980201197211831829922014328, 4.98204825486442364240106106447, 5.62852778629864333793496030835, 6.76328826156422739602668060433, 6.81282884808766437212669574833, 7.906627606183617297685905366073
