| L(s) = 1 | − 0.917·2-s − 1.10·3-s − 1.15·4-s + 1.01·6-s − 1.87·7-s + 2.89·8-s − 1.77·9-s + 0.565·11-s + 1.28·12-s + 0.286·13-s + 1.71·14-s − 0.340·16-s − 0.408·17-s + 1.62·18-s − 4.88·19-s + 2.07·21-s − 0.518·22-s + 1.65·23-s − 3.20·24-s − 0.262·26-s + 5.28·27-s + 2.16·28-s − 1.05·29-s + 0.534·31-s − 5.48·32-s − 0.625·33-s + 0.374·34-s + ⋯ |
| L(s) = 1 | − 0.648·2-s − 0.638·3-s − 0.579·4-s + 0.414·6-s − 0.707·7-s + 1.02·8-s − 0.591·9-s + 0.170·11-s + 0.370·12-s + 0.0793·13-s + 0.459·14-s − 0.0850·16-s − 0.0990·17-s + 0.383·18-s − 1.12·19-s + 0.452·21-s − 0.110·22-s + 0.344·23-s − 0.654·24-s − 0.0514·26-s + 1.01·27-s + 0.410·28-s − 0.196·29-s + 0.0959·31-s − 0.969·32-s − 0.108·33-s + 0.0642·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3366521929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3366521929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 0.917T + 2T^{2} \) |
| 3 | \( 1 + 1.10T + 3T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 - 0.565T + 11T^{2} \) |
| 13 | \( 1 - 0.286T + 13T^{2} \) |
| 17 | \( 1 + 0.408T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 31 | \( 1 - 0.534T + 31T^{2} \) |
| 37 | \( 1 + 0.861T + 37T^{2} \) |
| 41 | \( 1 + 4.66T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 - 4.36T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 - 6.97T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 5.35T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.482159157141913738562054416214, −8.102167376483913681493353198809, −6.87497350974939055702360730350, −6.46903149645437679141018683141, −5.49091035680655903525749980773, −4.84853713863815592315835822236, −3.92983018906762927255398093892, −3.01686708918259271755726599870, −1.69771349160729435942613200642, −0.38505986505971662066628408269,
0.38505986505971662066628408269, 1.69771349160729435942613200642, 3.01686708918259271755726599870, 3.92983018906762927255398093892, 4.84853713863815592315835822236, 5.49091035680655903525749980773, 6.46903149645437679141018683141, 6.87497350974939055702360730350, 8.102167376483913681493353198809, 8.482159157141913738562054416214
