| L(s) = 1 | + 0.161·2-s − 1.97·4-s + 5.15·7-s − 0.641·8-s + 1.28·11-s + 1.53·13-s + 0.832·14-s + 3.84·16-s + 2.43·17-s + 6.13·19-s + 0.207·22-s + 23-s + 0.247·26-s − 10.1·28-s + 6.10·29-s + 0.248·31-s + 1.90·32-s + 0.392·34-s − 1.29·37-s + 0.991·38-s − 5.44·41-s − 5.64·43-s − 2.53·44-s + 0.161·46-s + 7.37·47-s + 19.5·49-s − 3.02·52-s + ⋯ |
| L(s) = 1 | + 0.114·2-s − 0.986·4-s + 1.94·7-s − 0.226·8-s + 0.387·11-s + 0.424·13-s + 0.222·14-s + 0.961·16-s + 0.590·17-s + 1.40·19-s + 0.0442·22-s + 0.208·23-s + 0.0485·26-s − 1.92·28-s + 1.13·29-s + 0.0445·31-s + 0.336·32-s + 0.0673·34-s − 0.213·37-s + 0.160·38-s − 0.849·41-s − 0.861·43-s − 0.382·44-s + 0.0238·46-s + 1.07·47-s + 2.79·49-s − 0.419·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.487458263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.487458263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.161T + 2T^{2} \) |
| 7 | \( 1 - 5.15T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 - 0.248T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 + 7.36T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 5.51T + 73T^{2} \) |
| 79 | \( 1 - 0.0974T + 79T^{2} \) |
| 83 | \( 1 + 8.84T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 9.68T + 97T^{2} \) |
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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
−8.180571160789384477489724100609, −7.78291819230239062596326243141, −6.87256200835602930706097462469, −5.73641157115380474297566311279, −5.16805918221033122960261224255, −4.65984663596363344337023922391, −3.87786113472940641447052278137, −2.99330207757845276578569083605, −1.58807710115916272239700248441, −0.970801217109611142879721505640,
0.970801217109611142879721505640, 1.58807710115916272239700248441, 2.99330207757845276578569083605, 3.87786113472940641447052278137, 4.65984663596363344337023922391, 5.16805918221033122960261224255, 5.73641157115380474297566311279, 6.87256200835602930706097462469, 7.78291819230239062596326243141, 8.180571160789384477489724100609
