| L(s) = 1 | + 4.23·2-s + 9.94·4-s + 5·5-s − 7·7-s + 8.23·8-s + 21.1·10-s + 41.5·11-s + 88.9·13-s − 29.6·14-s − 44.6·16-s + 120.·17-s − 112.·19-s + 49.7·20-s + 175.·22-s + 115.·23-s + 25·25-s + 376.·26-s − 69.6·28-s + 144.·29-s − 258.·31-s − 255.·32-s + 509.·34-s − 35·35-s + 48.3·37-s − 475.·38-s + 41.1·40-s − 200.·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.24·4-s + 0.447·5-s − 0.377·7-s + 0.363·8-s + 0.669·10-s + 1.13·11-s + 1.89·13-s − 0.566·14-s − 0.697·16-s + 1.71·17-s − 1.35·19-s + 0.555·20-s + 1.70·22-s + 1.04·23-s + 0.200·25-s + 2.84·26-s − 0.469·28-s + 0.927·29-s − 1.49·31-s − 1.40·32-s + 2.57·34-s − 0.169·35-s + 0.214·37-s − 2.02·38-s + 0.162·40-s − 0.765·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.752723443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.752723443\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.23T + 8T^{2} \) |
| 11 | \( 1 - 41.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 88.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 258.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 48.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 575.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 184.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 151.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 529.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.28T + 3.00e5T^{2} \) |
| 71 | \( 1 - 61.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 484.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 878.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 491.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 415.T + 7.04e5T^{2} \) |
| 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
−11.47358152864396177753547895408, −10.61647835406717912417548105543, −9.345759010336154905613799357586, −8.431989026082797058890740456587, −6.71172219518177827087069410759, −6.20556051782324203043552656359, −5.23026954342136069670380104025, −3.90168472031826049292867180971, −3.23124910964062948756853364696, −1.45943654222981331992284209107,
1.45943654222981331992284209107, 3.23124910964062948756853364696, 3.90168472031826049292867180971, 5.23026954342136069670380104025, 6.20556051782324203043552656359, 6.71172219518177827087069410759, 8.431989026082797058890740456587, 9.345759010336154905613799357586, 10.61647835406717912417548105543, 11.47358152864396177753547895408
