| L(s) = 1 | + 4·2-s − 17.3·3-s + 16·4-s − 25·5-s − 69.3·6-s − 200.·7-s + 64·8-s + 57.3·9-s − 100·10-s − 277.·11-s − 277.·12-s + 655.·13-s − 802.·14-s + 433.·15-s + 256·16-s − 1.91e3·17-s + 229.·18-s + 910.·19-s − 400·20-s + 3.47e3·21-s − 1.11e3·22-s − 529·23-s − 1.10e3·24-s + 625·25-s + 2.62e3·26-s + 3.21e3·27-s − 3.21e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.11·3-s + 0.5·4-s − 0.447·5-s − 0.786·6-s − 1.54·7-s + 0.353·8-s + 0.236·9-s − 0.316·10-s − 0.691·11-s − 0.555·12-s + 1.07·13-s − 1.09·14-s + 0.497·15-s + 0.250·16-s − 1.60·17-s + 0.166·18-s + 0.578·19-s − 0.223·20-s + 1.72·21-s − 0.489·22-s − 0.208·23-s − 0.393·24-s + 0.200·25-s + 0.760·26-s + 0.849·27-s − 0.773·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.062138018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062138018\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 + 25T \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 + 17.3T + 243T^{2} \) |
| 7 | \( 1 + 200.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 277.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 655.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.91e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 910.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 1.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.16e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.27e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.64e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.92e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e5T + 8.58e9T^{2} \) |
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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
−11.37005968294403209196237385948, −10.76009956355073690029303713143, −9.638567639112689667261853764646, −8.256530791557163189229750714588, −6.70028863254781955151572851504, −6.29569029250365377781179911616, −5.17729079214960136342864349836, −3.93942369629795671951601422552, −2.75877412943828530738071895836, −0.55957276016202614190559618792,
0.55957276016202614190559618792, 2.75877412943828530738071895836, 3.93942369629795671951601422552, 5.17729079214960136342864349836, 6.29569029250365377781179911616, 6.70028863254781955151572851504, 8.256530791557163189229750714588, 9.638567639112689667261853764646, 10.76009956355073690029303713143, 11.37005968294403209196237385948
