L(s) = 1 | + 1.48·2-s − 0.193·3-s + 0.193·4-s − 0.518·5-s − 0.287·6-s + 0.806·7-s − 2.67·8-s − 2.96·9-s − 0.768·10-s + 0.156·11-s − 0.0376·12-s + 13-s + 1.19·14-s + 0.100·15-s − 4.35·16-s + 0.612·17-s − 4.38·18-s + 3.63·19-s − 0.100·20-s − 0.156·21-s + 0.231·22-s − 9.27·23-s + 0.518·24-s − 4.73·25-s + 1.48·26-s + 1.15·27-s + 0.156·28-s + ⋯ |
L(s) = 1 | + 1.04·2-s − 0.111·3-s + 0.0969·4-s − 0.232·5-s − 0.117·6-s + 0.304·7-s − 0.945·8-s − 0.987·9-s − 0.243·10-s + 0.0471·11-s − 0.0108·12-s + 0.277·13-s + 0.319·14-s + 0.0259·15-s − 1.08·16-s + 0.148·17-s − 1.03·18-s + 0.834·19-s − 0.0224·20-s − 0.0341·21-s + 0.0493·22-s − 1.93·23-s + 0.105·24-s − 0.946·25-s + 0.290·26-s + 0.222·27-s + 0.0295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 3 | \( 1 + 0.193T + 3T^{2} \) |
| 5 | \( 1 + 0.518T + 5T^{2} \) |
| 7 | \( 1 - 0.806T + 7T^{2} \) |
| 11 | \( 1 - 0.156T + 11T^{2} \) |
| 17 | \( 1 - 0.612T + 17T^{2} \) |
| 19 | \( 1 - 3.63T + 19T^{2} \) |
| 23 | \( 1 + 9.27T + 23T^{2} \) |
| 29 | \( 1 + 5.80T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 + 0.962T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 + 7.76T + 53T^{2} \) |
| 59 | \( 1 - 8.75T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 - 9.92T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 8.88T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.240939708144401226372143983063, −8.296662495564504086450322323302, −7.65020689832343175860820575071, −6.34831552855298886747637381296, −5.70164607445409131919060996190, −5.06858161784997611525524788325, −3.92945465276919834104895503363, −3.37123226052790036756153596399, −2.05931018707618849463116477124, 0,
2.05931018707618849463116477124, 3.37123226052790036756153596399, 3.92945465276919834104895503363, 5.06858161784997611525524788325, 5.70164607445409131919060996190, 6.34831552855298886747637381296, 7.65020689832343175860820575071, 8.296662495564504086450322323302, 9.240939708144401226372143983063