| L(s) = 1 | + 2.73·2-s − 3-s + 5.49·4-s − 1.98·5-s − 2.73·6-s − 2.67·7-s + 9.55·8-s + 9-s − 5.42·10-s − 5.81·11-s − 5.49·12-s + 5.48·13-s − 7.31·14-s + 1.98·15-s + 15.1·16-s + 17-s + 2.73·18-s − 8.60·19-s − 10.8·20-s + 2.67·21-s − 15.9·22-s − 5.12·23-s − 9.55·24-s − 1.07·25-s + 15.0·26-s − 27-s − 14.6·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.577·3-s + 2.74·4-s − 0.886·5-s − 1.11·6-s − 1.01·7-s + 3.37·8-s + 0.333·9-s − 1.71·10-s − 1.75·11-s − 1.58·12-s + 1.52·13-s − 1.95·14-s + 0.511·15-s + 3.79·16-s + 0.242·17-s + 0.645·18-s − 1.97·19-s − 2.43·20-s + 0.583·21-s − 3.39·22-s − 1.06·23-s − 1.95·24-s − 0.214·25-s + 2.94·26-s − 0.192·27-s − 2.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 19 | \( 1 + 8.60T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 - 5.45T + 29T^{2} \) |
| 31 | \( 1 - 2.02T + 31T^{2} \) |
| 37 | \( 1 - 0.206T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 + 1.74T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 4.58T + 71T^{2} \) |
| 73 | \( 1 - 0.210T + 73T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.86T + 89T^{2} \) |
| 97 | \( 1 - 0.435T + 97T^{2} \) |
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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
−7.973707358110139340997840002438, −6.90052851544560186876235912350, −6.22550682354958266915974158165, −5.97271979600013805574408802288, −4.91445936992508481660380186602, −4.32950835613221989361586773595, −3.55298676283555928235042135368, −2.96492711378821801960293895196, −1.86523894912335587858032679285, 0,
1.86523894912335587858032679285, 2.96492711378821801960293895196, 3.55298676283555928235042135368, 4.32950835613221989361586773595, 4.91445936992508481660380186602, 5.97271979600013805574408802288, 6.22550682354958266915974158165, 6.90052851544560186876235912350, 7.973707358110139340997840002438
