| L(s) = 1 | − 4.75·2-s + 3·3-s + 14.6·4-s + 7.60·5-s − 14.2·6-s − 25.1·7-s − 31.5·8-s + 9·9-s − 36.1·10-s + 11·11-s + 43.8·12-s − 41.9·13-s + 119.·14-s + 22.8·15-s + 32.9·16-s − 84.9·17-s − 42.8·18-s + 57.7·19-s + 111.·20-s − 75.4·21-s − 52.3·22-s + 125.·23-s − 94.5·24-s − 67.1·25-s + 199.·26-s + 27·27-s − 367.·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.82·4-s + 0.680·5-s − 0.970·6-s − 1.35·7-s − 1.39·8-s + 0.333·9-s − 1.14·10-s + 0.301·11-s + 1.05·12-s − 0.895·13-s + 2.28·14-s + 0.392·15-s + 0.514·16-s − 1.21·17-s − 0.560·18-s + 0.697·19-s + 1.24·20-s − 0.783·21-s − 0.507·22-s + 1.13·23-s − 0.804·24-s − 0.537·25-s + 1.50·26-s + 0.192·27-s − 2.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 4.75T + 8T^{2} \) |
| 5 | \( 1 - 7.60T + 125T^{2} \) |
| 7 | \( 1 + 25.1T + 343T^{2} \) |
| 13 | \( 1 + 41.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 53.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 184.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 353.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 147.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 700.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 589.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 141.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 437.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 598.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 393.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 721.T + 9.12e5T^{2} \) |
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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
−8.799903580090623174905767832121, −7.70396570538744643358629002291, −6.97320450709614326842551222956, −6.56213519446196873819104834518, −5.46555120751536683826526165362, −4.07230114460165340464828472356, −2.79073359016330623092728588320, −2.29954288276430170956232564246, −1.07616482684055716957556024981, 0,
1.07616482684055716957556024981, 2.29954288276430170956232564246, 2.79073359016330623092728588320, 4.07230114460165340464828472356, 5.46555120751536683826526165362, 6.56213519446196873819104834518, 6.97320450709614326842551222956, 7.70396570538744643358629002291, 8.799903580090623174905767832121
