| L(s) = 1 | + 0.385·2-s − 2.93·3-s − 1.85·4-s + 5-s − 1.13·6-s − 1.22·7-s − 1.48·8-s + 5.58·9-s + 0.385·10-s − 11-s + 5.42·12-s + 5.05·13-s − 0.474·14-s − 2.93·15-s + 3.12·16-s − 2.59·17-s + 2.15·18-s + 1.63·19-s − 1.85·20-s + 3.60·21-s − 0.385·22-s − 6.12·23-s + 4.35·24-s + 25-s + 1.95·26-s − 7.58·27-s + 2.27·28-s + ⋯ |
| L(s) = 1 | + 0.272·2-s − 1.69·3-s − 0.925·4-s + 0.447·5-s − 0.461·6-s − 0.464·7-s − 0.525·8-s + 1.86·9-s + 0.122·10-s − 0.301·11-s + 1.56·12-s + 1.40·13-s − 0.126·14-s − 0.756·15-s + 0.782·16-s − 0.629·17-s + 0.508·18-s + 0.374·19-s − 0.413·20-s + 0.785·21-s − 0.0822·22-s − 1.27·23-s + 0.888·24-s + 0.200·25-s + 0.382·26-s − 1.46·27-s + 0.429·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6541197130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6541197130\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.385T + 2T^{2} \) |
| 3 | \( 1 + 2.93T + 3T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 13 | \( 1 - 5.05T + 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 - 1.63T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 - 2.64T + 29T^{2} \) |
| 31 | \( 1 + 5.22T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 + 8.93T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 + 3.93T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 4.43T + 71T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 - 4.19T + 83T^{2} \) |
| 89 | \( 1 + 2.84T + 89T^{2} \) |
| 97 | \( 1 + 1.30T + 97T^{2} \) |
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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
−8.514367855079644924322911343795, −7.63727401454466954884504554339, −6.45628788823356570592835665320, −6.16276282930588950830555889324, −5.55958151905911904694311982999, −4.79746924024622883903811435425, −4.13743370203731945064664235554, −3.23241988215077602359002348031, −1.62081781665281341913861767446, −0.50443739064664274309983432626,
0.50443739064664274309983432626, 1.62081781665281341913861767446, 3.23241988215077602359002348031, 4.13743370203731945064664235554, 4.79746924024622883903811435425, 5.55958151905911904694311982999, 6.16276282930588950830555889324, 6.45628788823356570592835665320, 7.63727401454466954884504554339, 8.514367855079644924322911343795
