| L(s) = 1 | − 1.21·2-s + 1.67·3-s − 0.532·4-s − 5-s − 2.03·6-s − 7-s + 3.06·8-s − 0.179·9-s + 1.21·10-s − 5.20·11-s − 0.893·12-s + 4.41·13-s + 1.21·14-s − 1.67·15-s − 2.65·16-s − 0.830·17-s + 0.217·18-s − 7.19·19-s + 0.532·20-s − 1.67·21-s + 6.30·22-s − 1.03·23-s + 5.15·24-s + 25-s − 5.34·26-s − 5.33·27-s + 0.532·28-s + ⋯ |
| L(s) = 1 | − 0.856·2-s + 0.969·3-s − 0.266·4-s − 0.447·5-s − 0.830·6-s − 0.377·7-s + 1.08·8-s − 0.0599·9-s + 0.383·10-s − 1.56·11-s − 0.258·12-s + 1.22·13-s + 0.323·14-s − 0.433·15-s − 0.662·16-s − 0.201·17-s + 0.0513·18-s − 1.65·19-s + 0.119·20-s − 0.366·21-s + 1.34·22-s − 0.215·23-s + 1.05·24-s + 0.200·25-s − 1.04·26-s − 1.02·27-s + 0.100·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6104053170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6104053170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 - 1.67T + 3T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 0.830T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 + 1.03T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 - 0.399T + 31T^{2} \) |
| 37 | \( 1 - 6.46T + 37T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 8.62T + 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 6.22T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 5.80T + 89T^{2} \) |
| 97 | \( 1 + 8.15T + 97T^{2} \) |
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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
−8.107333720175987098220527765185, −7.65747269367685036540135010368, −6.60289536960090079534769851964, −5.85764404303917410397032428981, −4.91992453564019730991868470347, −4.09335192501166766317382567095, −3.48457413864833873689458082718, −2.56260953677051624914504948138, −1.81344167820489148585881152642, −0.40487953511880449598322869889,
0.40487953511880449598322869889, 1.81344167820489148585881152642, 2.56260953677051624914504948138, 3.48457413864833873689458082718, 4.09335192501166766317382567095, 4.91992453564019730991868470347, 5.85764404303917410397032428981, 6.60289536960090079534769851964, 7.65747269367685036540135010368, 8.107333720175987098220527765185
