| L(s) = 1 | − 2.57·2-s − 2.53·3-s + 4.65·4-s + 5-s + 6.53·6-s + 2.87·7-s − 6.84·8-s + 3.41·9-s − 2.57·10-s + 11-s − 11.7·12-s − 3.24·13-s − 7.42·14-s − 2.53·15-s + 8.35·16-s − 2.18·17-s − 8.80·18-s − 19-s + 4.65·20-s − 7.28·21-s − 2.57·22-s − 4.80·23-s + 17.3·24-s + 25-s + 8.37·26-s − 1.05·27-s + 13.3·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 1.46·3-s + 2.32·4-s + 0.447·5-s + 2.66·6-s + 1.08·7-s − 2.42·8-s + 1.13·9-s − 0.815·10-s + 0.301·11-s − 3.40·12-s − 0.900·13-s − 1.98·14-s − 0.653·15-s + 2.08·16-s − 0.530·17-s − 2.07·18-s − 0.229·19-s + 1.04·20-s − 1.59·21-s − 0.549·22-s − 1.00·23-s + 3.53·24-s + 0.200·25-s + 1.64·26-s − 0.202·27-s + 2.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 2.53T + 3T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 13 | \( 1 + 3.24T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 + 7.17T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 3.55T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 + 4.31T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 + 0.644T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 0.802T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 5.51T + 73T^{2} \) |
| 79 | \( 1 + 4.11T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 + 0.707T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−9.715655793122945541472172319866, −8.733618724765688512128365689384, −7.905202756884809565588741934772, −7.10143384156667570413327527521, −6.32111493888378534316284956818, −5.51328284543578083255431376033, −4.47772111267987667384104011234, −2.31837473866342331645629333156, −1.36447738691658004200208870611, 0,
1.36447738691658004200208870611, 2.31837473866342331645629333156, 4.47772111267987667384104011234, 5.51328284543578083255431376033, 6.32111493888378534316284956818, 7.10143384156667570413327527521, 7.905202756884809565588741934772, 8.733618724765688512128365689384, 9.715655793122945541472172319866
