| L(s) = 1 | − 2.98·2-s + 3·3-s + 0.905·4-s + 18.2·5-s − 8.95·6-s − 1.48·7-s + 21.1·8-s + 9·9-s − 54.5·10-s − 11·11-s + 2.71·12-s + 41.2·13-s + 4.42·14-s + 54.8·15-s − 70.4·16-s + 70.8·17-s − 26.8·18-s + 56.6·19-s + 16.5·20-s − 4.44·21-s + 32.8·22-s + 135.·23-s + 63.5·24-s + 209.·25-s − 123.·26-s + 27·27-s − 1.34·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.577·3-s + 0.113·4-s + 1.63·5-s − 0.609·6-s − 0.0800·7-s + 0.935·8-s + 0.333·9-s − 1.72·10-s − 0.301·11-s + 0.0653·12-s + 0.879·13-s + 0.0845·14-s + 0.943·15-s − 1.10·16-s + 1.01·17-s − 0.351·18-s + 0.683·19-s + 0.185·20-s − 0.0462·21-s + 0.318·22-s + 1.22·23-s + 0.540·24-s + 1.67·25-s − 0.927·26-s + 0.192·27-s − 0.00906·28-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
| \(L(2)\) |
\(\approx\) |
\(2.508204372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.508204372\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.98T + 8T^{2} \) |
| 5 | \( 1 - 18.2T + 125T^{2} \) |
| 7 | \( 1 + 1.48T + 343T^{2} \) |
| 13 | \( 1 - 41.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 56.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 135.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 17.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 27.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 4.16T + 7.95e4T^{2} \) |
| 47 | \( 1 + 26.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 155.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 67.6T + 2.05e5T^{2} \) |
| 67 | \( 1 - 624.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 693.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 258.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 511.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 971.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 61.7T + 9.12e5T^{2} \) |
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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
−8.933633211836720961612196672517, −8.252067977075792256342511989562, −7.44597584893875795855283111728, −6.60312458980415622910329825288, −5.62499695281887101164027029541, −4.96560468350412064642010364363, −3.61497013844574156621677805555, −2.58947462361283251619467451074, −1.57435155650379870463709817199, −0.929555929840361783726789141904,
0.929555929840361783726789141904, 1.57435155650379870463709817199, 2.58947462361283251619467451074, 3.61497013844574156621677805555, 4.96560468350412064642010364363, 5.62499695281887101164027029541, 6.60312458980415622910329825288, 7.44597584893875795855283111728, 8.252067977075792256342511989562, 8.933633211836720961612196672517
