| L(s) = 1 | − 3.27·2-s + 2.75·4-s − 5·5-s − 33.3·7-s + 17.2·8-s + 16.3·10-s + 11·11-s − 24.2·13-s + 109.·14-s − 78.4·16-s − 69.7·17-s − 125.·19-s − 13.7·20-s − 36.0·22-s − 130.·23-s + 25·25-s + 79.6·26-s − 91.8·28-s + 238.·29-s − 133.·31-s + 119.·32-s + 228.·34-s + 166.·35-s + 166.·37-s + 412.·38-s − 86.0·40-s − 297.·41-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.344·4-s − 0.447·5-s − 1.79·7-s + 0.760·8-s + 0.518·10-s + 0.301·11-s − 0.518·13-s + 2.08·14-s − 1.22·16-s − 0.994·17-s − 1.51·19-s − 0.153·20-s − 0.349·22-s − 1.18·23-s + 0.200·25-s + 0.600·26-s − 0.619·28-s + 1.52·29-s − 0.774·31-s + 0.661·32-s + 1.15·34-s + 0.804·35-s + 0.739·37-s + 1.76·38-s − 0.339·40-s − 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1917487187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1917487187\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3.27T + 8T^{2} \) |
| 7 | \( 1 + 33.3T + 343T^{2} \) |
| 13 | \( 1 + 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 69.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 238.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 166.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 585.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 40.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 391.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 368.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 877.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 9.99e2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.30e3T + 9.12e5T^{2} \) |
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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
−10.13208725473844605889578933583, −9.747275675129717312808326794674, −8.761845553401658604793617615118, −8.130080117004796774815452279924, −6.79520640485609957046133557291, −6.46622265580960318176740135212, −4.65793768807391151564811255336, −3.59950968056127728647506910416, −2.16731479193780361279616415094, −0.30586831223811438708661818051,
0.30586831223811438708661818051, 2.16731479193780361279616415094, 3.59950968056127728647506910416, 4.65793768807391151564811255336, 6.46622265580960318176740135212, 6.79520640485609957046133557291, 8.130080117004796774815452279924, 8.761845553401658604793617615118, 9.747275675129717312808326794674, 10.13208725473844605889578933583
