| L(s) = 1 | − 2.32·2-s + 3-s + 3.41·4-s − 3.48·5-s − 2.32·6-s + 3.88·7-s − 3.29·8-s + 9-s + 8.09·10-s − 3.96·11-s + 3.41·12-s + 0.505·13-s − 9.03·14-s − 3.48·15-s + 0.828·16-s − 17-s − 2.32·18-s − 2.84·19-s − 11.8·20-s + 3.88·21-s + 9.22·22-s − 1.37·23-s − 3.29·24-s + 7.11·25-s − 1.17·26-s + 27-s + 13.2·28-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 0.577·3-s + 1.70·4-s − 1.55·5-s − 0.949·6-s + 1.46·7-s − 1.16·8-s + 0.333·9-s + 2.56·10-s − 1.19·11-s + 0.985·12-s + 0.140·13-s − 2.41·14-s − 0.898·15-s + 0.207·16-s − 0.242·17-s − 0.548·18-s − 0.652·19-s − 2.65·20-s + 0.847·21-s + 1.96·22-s − 0.287·23-s − 0.671·24-s + 1.42·25-s − 0.230·26-s + 0.192·27-s + 2.50·28-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 | \( 1 + 2.32T + 2T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 + 3.96T + 11T^{2} \) |
| 13 | \( 1 - 0.505T + 13T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 - 4.97T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 - 4.01T + 41T^{2} \) |
| 43 | \( 1 - 3.11T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 8.83T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 83 | \( 1 + 3.56T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.134154047570051653164329122265, −7.67884332444507458727644958396, −7.31125384776411561836549175805, −6.19665056077953871859210099740, −4.72722584318792799040029192848, −4.40528907576227089036547382887, −3.07511808388795468463774913553, −2.22779819394519362931562816803, −1.17720758602190202532631430209, 0,
1.17720758602190202532631430209, 2.22779819394519362931562816803, 3.07511808388795468463774913553, 4.40528907576227089036547382887, 4.72722584318792799040029192848, 6.19665056077953871859210099740, 7.31125384776411561836549175805, 7.67884332444507458727644958396, 8.134154047570051653164329122265
