| L(s) = 1 | − 1.61·2-s + 3.23·3-s + 0.618·4-s − 5.23·6-s − 0.236·7-s + 2.23·8-s + 7.47·9-s + 2·11-s + 2.00·12-s + 3.23·13-s + 0.381·14-s − 4.85·16-s − 0.763·17-s − 12.0·18-s − 2.23·19-s − 0.763·21-s − 3.23·22-s − 5.70·23-s + 7.23·24-s − 5.23·26-s + 14.4·27-s − 0.145·28-s + 2.76·29-s + 31-s + 3.38·32-s + 6.47·33-s + 1.23·34-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 1.86·3-s + 0.309·4-s − 2.13·6-s − 0.0892·7-s + 0.790·8-s + 2.49·9-s + 0.603·11-s + 0.577·12-s + 0.897·13-s + 0.102·14-s − 1.21·16-s − 0.185·17-s − 2.84·18-s − 0.512·19-s − 0.166·21-s − 0.689·22-s − 1.19·23-s + 1.47·24-s − 1.02·26-s + 2.78·27-s − 0.0275·28-s + 0.513·29-s + 0.179·31-s + 0.597·32-s + 1.12·33-s + 0.211·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668486609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668486609\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 2.76T + 29T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 8.18T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−9.967280589158189204285792524518, −9.228865571499554820667182546302, −8.644575531911551606064562873990, −8.121435168221907132844902607521, −7.33870137060633526820861246616, −6.34991273284939083688759986235, −4.41908969131775492640818384608, −3.71969600517394134215624020018, −2.39402175209344587667865896619, −1.34706098564617273358026624737,
1.34706098564617273358026624737, 2.39402175209344587667865896619, 3.71969600517394134215624020018, 4.41908969131775492640818384608, 6.34991273284939083688759986235, 7.33870137060633526820861246616, 8.121435168221907132844902607521, 8.644575531911551606064562873990, 9.228865571499554820667182546302, 9.967280589158189204285792524518
