| L(s) = 1 | − 2.73·2-s − 0.544·4-s − 10.5·5-s + 15.6·7-s + 23.3·8-s + 28.9·10-s + 0.571·11-s − 88.1·13-s − 42.7·14-s − 59.3·16-s + 61.0·17-s − 111.·19-s + 5.77·20-s − 1.56·22-s − 169.·23-s − 12.6·25-s + 240.·26-s − 8.53·28-s + 68.0·29-s + 144.·31-s − 24.6·32-s − 166.·34-s − 166.·35-s − 21.1·37-s + 305.·38-s − 247.·40-s + 91.9·41-s + ⋯ |
| L(s) = 1 | − 0.965·2-s − 0.0681·4-s − 0.948·5-s + 0.845·7-s + 1.03·8-s + 0.915·10-s + 0.0156·11-s − 1.88·13-s − 0.816·14-s − 0.927·16-s + 0.870·17-s − 1.35·19-s + 0.0645·20-s − 0.0151·22-s − 1.53·23-s − 0.101·25-s + 1.81·26-s − 0.0576·28-s + 0.435·29-s + 0.834·31-s − 0.135·32-s − 0.840·34-s − 0.801·35-s − 0.0938·37-s + 1.30·38-s − 0.977·40-s + 0.350·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3467729779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3467729779\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 2.73T + 8T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 - 15.6T + 343T^{2} \) |
| 11 | \( 1 - 0.571T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 61.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 68.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 21.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 91.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 234.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 82.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 110.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 16.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 875.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 53.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 21.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 598.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 995.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.13e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 92.2T + 9.12e5T^{2} \) |
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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
−8.533014043984425749215191891991, −7.86764400941138362856515740943, −7.72421574849504860507140784511, −6.70712549603182317515254697568, −5.43120575438700009549394594086, −4.49737073457986294861572196612, −4.11692975052619584627372578251, −2.58871090704901683639611517464, −1.59510641248871215508298290535, −0.31413306203625851922630434190,
0.31413306203625851922630434190, 1.59510641248871215508298290535, 2.58871090704901683639611517464, 4.11692975052619584627372578251, 4.49737073457986294861572196612, 5.43120575438700009549394594086, 6.70712549603182317515254697568, 7.72421574849504860507140784511, 7.86764400941138362856515740943, 8.533014043984425749215191891991
