| L(s) = 1 | − 1.09·2-s − 2.94·3-s − 0.800·4-s + 5-s + 3.22·6-s + 2.47·7-s + 3.06·8-s + 5.67·9-s − 1.09·10-s + 11-s + 2.35·12-s + 0.870·13-s − 2.70·14-s − 2.94·15-s − 1.75·16-s + 5.55·17-s − 6.21·18-s + 0.757·19-s − 0.800·20-s − 7.28·21-s − 1.09·22-s − 7.95·23-s − 9.03·24-s + 25-s − 0.953·26-s − 7.88·27-s − 1.97·28-s + ⋯ |
| L(s) = 1 | − 0.774·2-s − 1.70·3-s − 0.400·4-s + 0.447·5-s + 1.31·6-s + 0.934·7-s + 1.08·8-s + 1.89·9-s − 0.346·10-s + 0.301·11-s + 0.681·12-s + 0.241·13-s − 0.723·14-s − 0.760·15-s − 0.439·16-s + 1.34·17-s − 1.46·18-s + 0.173·19-s − 0.179·20-s − 1.58·21-s − 0.233·22-s − 1.65·23-s − 1.84·24-s + 0.200·25-s − 0.186·26-s − 1.51·27-s − 0.374·28-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 | \( 1 + 1.09T + 2T^{2} \) |
| 3 | \( 1 + 2.94T + 3T^{2} \) |
| 7 | \( 1 - 2.47T + 7T^{2} \) |
| 13 | \( 1 - 0.870T + 13T^{2} \) |
| 17 | \( 1 - 5.55T + 17T^{2} \) |
| 19 | \( 1 - 0.757T + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 6.68T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 - 6.03T + 59T^{2} \) |
| 61 | \( 1 - 3.86T + 61T^{2} \) |
| 67 | \( 1 - 1.40T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 7.91T + 97T^{2} \) |
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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
−8.070021797013921366142978049162, −7.42241585023066535495724080807, −6.56524338920161498244986392613, −5.70220979714542553467856758923, −5.26886718591201168213869922933, −4.54056978947934335847376071070, −3.69657694579481068949410934939, −1.76395626479941886094266999203, −1.20589639766477457603908917639, 0,
1.20589639766477457603908917639, 1.76395626479941886094266999203, 3.69657694579481068949410934939, 4.54056978947934335847376071070, 5.26886718591201168213869922933, 5.70220979714542553467856758923, 6.56524338920161498244986392613, 7.42241585023066535495724080807, 8.070021797013921366142978049162
