| L(s) = 1 | − 1.10·2-s − 6.78·4-s − 20.8·5-s − 33.6·7-s + 16.3·8-s + 23.0·10-s − 69.0·11-s − 37.7·13-s + 37.1·14-s + 36.2·16-s − 83.4·17-s − 80.0·19-s + 141.·20-s + 76.2·22-s − 1.95·23-s + 310.·25-s + 41.6·26-s + 228.·28-s + 169.·29-s + 251.·31-s − 170.·32-s + 92.0·34-s + 701.·35-s + 13.1·37-s + 88.3·38-s − 340.·40-s − 150.·41-s + ⋯ |
| L(s) = 1 | − 0.390·2-s − 0.847·4-s − 1.86·5-s − 1.81·7-s + 0.721·8-s + 0.728·10-s − 1.89·11-s − 0.804·13-s + 0.708·14-s + 0.566·16-s − 1.19·17-s − 0.966·19-s + 1.58·20-s + 0.738·22-s − 0.0177·23-s + 2.48·25-s + 0.314·26-s + 1.53·28-s + 1.08·29-s + 1.45·31-s − 0.942·32-s + 0.464·34-s + 3.38·35-s + 0.0582·37-s + 0.377·38-s − 1.34·40-s − 0.574·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 1.10T + 8T^{2} \) |
| 5 | \( 1 + 20.8T + 125T^{2} \) |
| 7 | \( 1 + 33.6T + 343T^{2} \) |
| 11 | \( 1 + 69.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 83.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.95T + 1.21e4T^{2} \) |
| 29 | \( 1 - 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 335.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 457.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 207.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 273.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 729.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 385.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 351.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 537.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 486.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 17.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−8.377393828770551726647111713147, −7.67865338507201570338087089745, −7.03660200337702035282321102197, −6.10998932755068543283473469438, −4.62153123867813686093388618459, −4.52185782211754289084438012675, −3.25852084585917644786660298287, −2.70996072941532142580932352863, −0.43626017435466222875802167370, 0,
0.43626017435466222875802167370, 2.70996072941532142580932352863, 3.25852084585917644786660298287, 4.52185782211754289084438012675, 4.62153123867813686093388618459, 6.10998932755068543283473469438, 7.03660200337702035282321102197, 7.67865338507201570338087089745, 8.377393828770551726647111713147
