| L(s) = 1 | + 1.82·2-s + 3·3-s − 4.65·4-s + 5·5-s + 5.48·6-s − 15.0·7-s − 23.1·8-s + 9·9-s + 9.13·10-s − 13.9·12-s − 11.4·13-s − 27.5·14-s + 15·15-s − 5.02·16-s + 11.9·17-s + 16.4·18-s − 91.0·19-s − 23.2·20-s − 45.2·21-s + 103.·23-s − 69.4·24-s + 25·25-s − 20.9·26-s + 27·27-s + 70.3·28-s − 99.3·29-s + 27.4·30-s + ⋯ |
| L(s) = 1 | + 0.646·2-s + 0.577·3-s − 0.582·4-s + 0.447·5-s + 0.373·6-s − 0.814·7-s − 1.02·8-s + 0.333·9-s + 0.289·10-s − 0.336·12-s − 0.243·13-s − 0.526·14-s + 0.258·15-s − 0.0785·16-s + 0.170·17-s + 0.215·18-s − 1.09·19-s − 0.260·20-s − 0.470·21-s + 0.935·23-s − 0.590·24-s + 0.200·25-s − 0.157·26-s + 0.192·27-s + 0.474·28-s − 0.636·29-s + 0.166·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.538388707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.538388707\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.82T + 8T^{2} \) |
| 7 | \( 1 + 15.0T + 343T^{2} \) |
| 13 | \( 1 + 11.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 39.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 304.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 552.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 251.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 211.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 947.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 175.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 768.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 710.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 902.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 482.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 724.T + 9.12e5T^{2} \) |
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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
−9.034494663235909601034781090223, −8.287183952016216942060179859744, −7.22073596198062163027277846322, −6.37786738690084876729241662466, −5.63508626577194001133394378793, −4.72214478772715437945328574305, −3.87414121501668631345885518995, −3.09247010436275058565313815379, −2.19244778740010749458144788440, −0.64395549302188950099032247497,
0.64395549302188950099032247497, 2.19244778740010749458144788440, 3.09247010436275058565313815379, 3.87414121501668631345885518995, 4.72214478772715437945328574305, 5.63508626577194001133394378793, 6.37786738690084876729241662466, 7.22073596198062163027277846322, 8.287183952016216942060179859744, 9.034494663235909601034781090223
