| L(s) = 1 | + 5.03·2-s + 17.3·4-s − 5·5-s + 18.5·7-s + 46.8·8-s − 25.1·10-s − 11·11-s + 47.0·13-s + 93.4·14-s + 97.2·16-s + 2.62·17-s + 157.·19-s − 86.5·20-s − 55.3·22-s − 186.·23-s + 25·25-s + 236.·26-s + 321.·28-s + 270.·29-s − 25.7·31-s + 114.·32-s + 13.2·34-s − 92.8·35-s + 228.·37-s + 791.·38-s − 234.·40-s − 6.85·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.16·4-s − 0.447·5-s + 1.00·7-s + 2.07·8-s − 0.795·10-s − 0.301·11-s + 1.00·13-s + 1.78·14-s + 1.51·16-s + 0.0374·17-s + 1.89·19-s − 0.967·20-s − 0.536·22-s − 1.69·23-s + 0.200·25-s + 1.78·26-s + 2.17·28-s + 1.73·29-s − 0.149·31-s + 0.632·32-s + 0.0666·34-s − 0.448·35-s + 1.01·37-s + 3.37·38-s − 0.926·40-s − 0.0261·41-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
| \(L(2)\) |
\(\approx\) |
\(6.306200598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.306200598\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.03T + 8T^{2} \) |
| 7 | \( 1 - 18.5T + 343T^{2} \) |
| 13 | \( 1 - 47.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.62T + 4.91e3T^{2} \) |
| 19 | \( 1 - 157.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 186.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 270.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 25.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 228.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.85T + 6.89e4T^{2} \) |
| 43 | \( 1 + 386.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 283.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 452.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 88.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 829.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 489.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 63.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 78.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 203.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 62.6T + 9.12e5T^{2} \) |
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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
−11.04094199873332141917923751956, −9.981835833721529354774837526248, −8.344681591283091443022760338387, −7.65918315221211126005799434008, −6.52934070117816720102452374788, −5.57695788151730533949348506145, −4.77007964444911084802163011730, −3.87243210216886785235112897481, −2.87505852019845068425624676981, −1.43620642953341971057236284464,
1.43620642953341971057236284464, 2.87505852019845068425624676981, 3.87243210216886785235112897481, 4.77007964444911084802163011730, 5.57695788151730533949348506145, 6.52934070117816720102452374788, 7.65918315221211126005799434008, 8.344681591283091443022760338387, 9.981835833721529354774837526248, 11.04094199873332141917923751956
