| L(s) = 1 | − 5.34·2-s − 7.64·3-s + 20.5·4-s − 5·5-s + 40.9·6-s − 21.7·7-s − 67.3·8-s + 31.5·9-s + 26.7·10-s + 30.4·11-s − 157.·12-s − 13·13-s + 116.·14-s + 38.2·15-s + 195.·16-s − 20.7·17-s − 168.·18-s − 36.7·19-s − 102.·20-s + 166.·21-s − 162.·22-s − 75.7·23-s + 515.·24-s + 25·25-s + 69.5·26-s − 34.4·27-s − 448.·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 1.47·3-s + 2.57·4-s − 0.447·5-s + 2.78·6-s − 1.17·7-s − 2.97·8-s + 1.16·9-s + 0.845·10-s + 0.834·11-s − 3.78·12-s − 0.277·13-s + 2.22·14-s + 0.658·15-s + 3.05·16-s − 0.295·17-s − 2.20·18-s − 0.443·19-s − 1.15·20-s + 1.73·21-s − 1.57·22-s − 0.686·23-s + 4.38·24-s + 0.200·25-s + 0.524·26-s − 0.245·27-s − 3.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 5.34T + 8T^{2} \) |
| 3 | \( 1 + 7.64T + 27T^{2} \) |
| 7 | \( 1 + 21.7T + 343T^{2} \) |
| 11 | \( 1 - 30.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 20.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 75.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 36.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 128.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 103.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 43.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 349.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 313.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 267.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 698.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 373.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 560.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 904.T + 9.12e5T^{2} \) |
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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
−8.573045151241399488481509373555, −7.53742128249219625142974589571, −6.89875214982578612892356795341, −6.30773187366419025296758910474, −5.76749728546430771455474840202, −4.29623864370109520782474391061, −3.10133540412868132154203176083, −1.80144781648110861850213785538, −0.65645252824297388206079964141, 0,
0.65645252824297388206079964141, 1.80144781648110861850213785538, 3.10133540412868132154203176083, 4.29623864370109520782474391061, 5.76749728546430771455474840202, 6.30773187366419025296758910474, 6.89875214982578612892356795341, 7.53742128249219625142974589571, 8.573045151241399488481509373555
