| L(s) = 1 | − 1.55·2-s + 0.350·3-s + 0.406·4-s − 2.80·5-s − 0.544·6-s − 3.32·7-s + 2.47·8-s − 2.87·9-s + 4.35·10-s − 3.82·11-s + 0.142·12-s + 0.682·13-s + 5.15·14-s − 0.983·15-s − 4.64·16-s + 2.91·17-s + 4.46·18-s + 4.20·19-s − 1.14·20-s − 1.16·21-s + 5.93·22-s + 2.71·23-s + 0.866·24-s + 2.87·25-s − 1.05·26-s − 2.06·27-s − 1.35·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.202·3-s + 0.203·4-s − 1.25·5-s − 0.222·6-s − 1.25·7-s + 0.873·8-s − 0.959·9-s + 1.37·10-s − 1.15·11-s + 0.0411·12-s + 0.189·13-s + 1.37·14-s − 0.254·15-s − 1.16·16-s + 0.706·17-s + 1.05·18-s + 0.964·19-s − 0.255·20-s − 0.254·21-s + 1.26·22-s + 0.566·23-s + 0.176·24-s + 0.574·25-s − 0.207·26-s − 0.396·27-s − 0.255·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 3 | \( 1 - 0.350T + 3T^{2} \) |
| 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 - 0.682T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 - 5.25T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 - 6.67T + 47T^{2} \) |
| 53 | \( 1 + 0.715T + 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 - 3.18T + 71T^{2} \) |
| 73 | \( 1 - 7.53T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 4.35T + 83T^{2} \) |
| 89 | \( 1 - 9.22T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−8.054385892668177128230023803241, −7.62778546667736746577612986809, −7.03632422126277969434591811572, −5.90419395969504321748546818410, −5.14703550223467924667379653175, −4.05467790615260832546377754655, −3.25940920946107893176179950227, −2.60053018705316162628391150008, −0.845916660000218914049776647738, 0,
0.845916660000218914049776647738, 2.60053018705316162628391150008, 3.25940920946107893176179950227, 4.05467790615260832546377754655, 5.14703550223467924667379653175, 5.90419395969504321748546818410, 7.03632422126277969434591811572, 7.62778546667736746577612986809, 8.054385892668177128230023803241
