| L(s) = 1 | − 0.692·2-s − 2.63·3-s − 1.52·4-s + 5-s + 1.82·6-s + 0.337·7-s + 2.43·8-s + 3.94·9-s − 0.692·10-s − 11-s + 4.00·12-s + 0.435·13-s − 0.233·14-s − 2.63·15-s + 1.35·16-s − 1.27·17-s − 2.73·18-s − 1.51·19-s − 1.52·20-s − 0.888·21-s + 0.692·22-s − 1.16·23-s − 6.42·24-s + 25-s − 0.301·26-s − 2.48·27-s − 0.512·28-s + ⋯ |
| L(s) = 1 | − 0.489·2-s − 1.52·3-s − 0.760·4-s + 0.447·5-s + 0.744·6-s + 0.127·7-s + 0.861·8-s + 1.31·9-s − 0.218·10-s − 0.301·11-s + 1.15·12-s + 0.120·13-s − 0.0623·14-s − 0.680·15-s + 0.338·16-s − 0.308·17-s − 0.643·18-s − 0.348·19-s − 0.340·20-s − 0.193·21-s + 0.147·22-s − 0.243·23-s − 1.31·24-s + 0.200·25-s − 0.0590·26-s − 0.479·27-s − 0.0968·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 0.692T + 2T^{2} \) |
| 3 | \( 1 + 2.63T + 3T^{2} \) |
| 7 | \( 1 - 0.337T + 7T^{2} \) |
| 13 | \( 1 - 0.435T + 13T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 + 4.07T + 31T^{2} \) |
| 37 | \( 1 + 7.02T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 6.91T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 - 2.17T + 53T^{2} \) |
| 59 | \( 1 - 9.68T + 59T^{2} \) |
| 61 | \( 1 + 5.63T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.203029819998620271680988982105, −7.23159735085556417079013013100, −6.57471705839419526401020030209, −5.71050067451664788345557582661, −5.20523185826434294302916445911, −4.55038282500007479791762409564, −3.65935449610626790001109895460, −2.10448500221651609387746484807, −1.01067395689286444005868274253, 0,
1.01067395689286444005868274253, 2.10448500221651609387746484807, 3.65935449610626790001109895460, 4.55038282500007479791762409564, 5.20523185826434294302916445911, 5.71050067451664788345557582661, 6.57471705839419526401020030209, 7.23159735085556417079013013100, 8.203029819998620271680988982105
