| L(s) = 1 | − 0.780·2-s − 7.39·4-s + 21.9·5-s + 32.1·7-s + 12.0·8-s − 17.0·10-s + 35.3·11-s + 34.5·13-s − 25.0·14-s + 49.7·16-s + 27.9·17-s + 74.3·19-s − 162.·20-s − 27.5·22-s − 128.·23-s + 355.·25-s − 26.9·26-s − 237.·28-s − 42.7·29-s − 211.·31-s − 134.·32-s − 21.8·34-s + 703.·35-s − 165.·37-s − 57.9·38-s + 263.·40-s − 377.·41-s + ⋯ |
| L(s) = 1 | − 0.275·2-s − 0.923·4-s + 1.96·5-s + 1.73·7-s + 0.530·8-s − 0.540·10-s + 0.967·11-s + 0.736·13-s − 0.478·14-s + 0.777·16-s + 0.399·17-s + 0.897·19-s − 1.81·20-s − 0.266·22-s − 1.16·23-s + 2.84·25-s − 0.203·26-s − 1.60·28-s − 0.273·29-s − 1.22·31-s − 0.745·32-s − 0.110·34-s + 3.39·35-s − 0.736·37-s − 0.247·38-s + 1.04·40-s − 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.785277828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.785277828\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 + 0.780T + 8T^{2} \) |
| 5 | \( 1 - 21.9T + 125T^{2} \) |
| 7 | \( 1 - 32.1T + 343T^{2} \) |
| 11 | \( 1 - 35.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 165.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 377.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 261.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 113.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 337.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 183.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 168.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 805.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 797.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 251.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 653.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.57e3T + 9.12e5T^{2} \) |
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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
−10.19964876830435832858166966784, −9.544577840648494619202415717171, −8.760025111983606357370570903924, −8.073418070076994855257099392047, −6.70284625091343505600466472607, −5.46847520484038470116076390409, −5.12851834159217484273405422634, −3.75440176809866945502079947770, −1.78153916726789082834019090019, −1.32625855242284174807916449563,
1.32625855242284174807916449563, 1.78153916726789082834019090019, 3.75440176809866945502079947770, 5.12851834159217484273405422634, 5.46847520484038470116076390409, 6.70284625091343505600466472607, 8.073418070076994855257099392047, 8.760025111983606357370570903924, 9.544577840648494619202415717171, 10.19964876830435832858166966784
