| L(s) = 1 | − 2.73·2-s + 5.46·4-s − 5-s + 0.267·7-s − 9.46·8-s + 2.73·10-s + 5.46·11-s − 13-s − 0.732·14-s + 14.9·16-s − 4·17-s + 6·19-s − 5.46·20-s − 14.9·22-s + 3·23-s + 25-s + 2.73·26-s + 1.46·28-s − 6.19·29-s − 2.73·31-s − 21.8·32-s + 10.9·34-s − 0.267·35-s + 9.19·37-s − 16.3·38-s + 9.46·40-s + 1.73·41-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.73·4-s − 0.447·5-s + 0.101·7-s − 3.34·8-s + 0.863·10-s + 1.64·11-s − 0.277·13-s − 0.195·14-s + 3.73·16-s − 0.970·17-s + 1.37·19-s − 1.22·20-s − 3.18·22-s + 0.625·23-s + 0.200·25-s + 0.535·26-s + 0.276·28-s − 1.15·29-s − 0.490·31-s − 3.86·32-s + 1.87·34-s − 0.0452·35-s + 1.51·37-s − 2.65·38-s + 1.49·40-s + 0.270·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6950488088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6950488088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 7 | \( 1 - 0.267T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 9.19T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 - 8.46T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 3.19T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 0.803T + 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 + 4.92T + 83T^{2} \) |
| 89 | \( 1 + 9.19T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.401932322248301028623769940855, −8.684979063000861826071002499440, −7.82842085718119201500461869624, −7.13609329059899322049092692920, −6.62008037625696676457415625697, −5.60315985735340452097850243363, −4.07972200396542977448512967144, −2.98917032980032608316669523648, −1.79147208016082063407051441578, −0.78526644600876922495914504547,
0.78526644600876922495914504547, 1.79147208016082063407051441578, 2.98917032980032608316669523648, 4.07972200396542977448512967144, 5.60315985735340452097850243363, 6.62008037625696676457415625697, 7.13609329059899322049092692920, 7.82842085718119201500461869624, 8.684979063000861826071002499440, 9.401932322248301028623769940855
