| L(s) = 1 | + 1.22·2-s − 1.28·3-s − 0.505·4-s − 1.57·6-s − 7-s − 3.06·8-s − 1.34·9-s − 1.83·11-s + 0.650·12-s − 3.24·13-s − 1.22·14-s − 2.73·16-s − 5.53·17-s − 1.64·18-s − 2.06·19-s + 1.28·21-s − 2.23·22-s − 23-s + 3.94·24-s − 3.96·26-s + 5.58·27-s + 0.505·28-s + 2.29·29-s − 2.49·31-s + 2.78·32-s + 2.35·33-s − 6.76·34-s + ⋯ |
| L(s) = 1 | + 0.864·2-s − 0.742·3-s − 0.252·4-s − 0.642·6-s − 0.377·7-s − 1.08·8-s − 0.448·9-s − 0.552·11-s + 0.187·12-s − 0.900·13-s − 0.326·14-s − 0.683·16-s − 1.34·17-s − 0.387·18-s − 0.474·19-s + 0.280·21-s − 0.477·22-s − 0.208·23-s + 0.804·24-s − 0.778·26-s + 1.07·27-s + 0.0956·28-s + 0.426·29-s − 0.447·31-s + 0.492·32-s + 0.410·33-s − 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6262959612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6262959612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 3 | \( 1 + 1.28T + 3T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 29 | \( 1 - 2.29T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 + 7.37T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 3.84T + 43T^{2} \) |
| 47 | \( 1 - 5.75T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 9.36T + 97T^{2} \) |
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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
−8.625400980481228539370194564158, −7.52483040493571781760379544425, −6.71056690553361219846606292848, −5.99138702828466039617463324747, −5.46810531370927918487210348120, −4.68060376117948058112051088649, −4.14990378728655759226689319273, −2.99612171369277443509683710111, −2.32473218274964269960898579433, −0.38442139542392850176277093901,
0.38442139542392850176277093901, 2.32473218274964269960898579433, 2.99612171369277443509683710111, 4.14990378728655759226689319273, 4.68060376117948058112051088649, 5.46810531370927918487210348120, 5.99138702828466039617463324747, 6.71056690553361219846606292848, 7.52483040493571781760379544425, 8.625400980481228539370194564158
