| L(s) = 1 | + 0.257·2-s + 3.28·3-s − 1.93·4-s + 0.843·6-s − 3.60·7-s − 1.01·8-s + 7.77·9-s − 1.43·11-s − 6.34·12-s − 2.88·13-s − 0.926·14-s + 3.60·16-s − 0.875·17-s + 1.99·18-s + 2.64·19-s − 11.8·21-s − 0.369·22-s + 4.70·23-s − 3.31·24-s − 0.741·26-s + 15.6·27-s + 6.97·28-s − 3.36·29-s − 0.807·31-s + 2.95·32-s − 4.72·33-s − 0.225·34-s + ⋯ |
| L(s) = 1 | + 0.181·2-s + 1.89·3-s − 0.966·4-s + 0.344·6-s − 1.36·7-s − 0.357·8-s + 2.59·9-s − 0.433·11-s − 1.83·12-s − 0.799·13-s − 0.247·14-s + 0.901·16-s − 0.212·17-s + 0.470·18-s + 0.606·19-s − 2.58·21-s − 0.0788·22-s + 0.980·23-s − 0.677·24-s − 0.145·26-s + 3.01·27-s + 1.31·28-s − 0.624·29-s − 0.145·31-s + 0.521·32-s − 0.822·33-s − 0.0386·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.257T + 2T^{2} \) |
| 3 | \( 1 - 3.28T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 + 0.875T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 4.70T + 23T^{2} \) |
| 29 | \( 1 + 3.36T + 29T^{2} \) |
| 31 | \( 1 + 0.807T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 + 9.22T + 41T^{2} \) |
| 43 | \( 1 - 4.15T + 43T^{2} \) |
| 47 | \( 1 + 1.63T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.00T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 0.971T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−7.76123034974798186156743668087, −7.27312755881137869949709150016, −6.50678526567613366090135667287, −5.36739531493157041765457423368, −4.61275451468161604856679481498, −3.81470557207170104669845900602, −3.10741373562538554533516847092, −2.80695072141997925700214337219, −1.53252821230364102447961612913, 0,
1.53252821230364102447961612913, 2.80695072141997925700214337219, 3.10741373562538554533516847092, 3.81470557207170104669845900602, 4.61275451468161604856679481498, 5.36739531493157041765457423368, 6.50678526567613366090135667287, 7.27312755881137869949709150016, 7.76123034974798186156743668087
