| L(s) = 1 | + 1.56·2-s − 5.56·4-s + 3.56·5-s − 27.1·7-s − 21.1·8-s + 5.56·10-s − 15.2·11-s − 13·13-s − 42.4·14-s + 11.4·16-s − 44.5·17-s + 23.9·19-s − 19.8·20-s − 23.8·22-s − 122.·23-s − 112.·25-s − 20.3·26-s + 151.·28-s + 219.·29-s + 27.0·31-s + 187.·32-s − 69.5·34-s − 96.7·35-s + 94.1·37-s + 37.4·38-s − 75.4·40-s + 160.·41-s + ⋯ |
| L(s) = 1 | + 0.552·2-s − 0.695·4-s + 0.318·5-s − 1.46·7-s − 0.935·8-s + 0.175·10-s − 0.418·11-s − 0.277·13-s − 0.810·14-s + 0.178·16-s − 0.635·17-s + 0.289·19-s − 0.221·20-s − 0.230·22-s − 1.11·23-s − 0.898·25-s − 0.153·26-s + 1.02·28-s + 1.40·29-s + 0.156·31-s + 1.03·32-s − 0.350·34-s − 0.467·35-s + 0.418·37-s + 0.159·38-s − 0.298·40-s + 0.610·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 1.56T + 8T^{2} \) |
| 5 | \( 1 - 3.56T + 125T^{2} \) |
| 7 | \( 1 + 27.1T + 343T^{2} \) |
| 11 | \( 1 + 15.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 27.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 160.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 466.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 120.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 512.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 586.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.51e3T + 9.12e5T^{2} \) |
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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
−12.82561550606998652171882579246, −11.83983175074435025619761270369, −10.09807116794461824291465021047, −9.561810872571921636712154273266, −8.292856303885674255776755181762, −6.61946123359073797781207247279, −5.64369465695127682206552627180, −4.20795793242617387384424485764, −2.86923266026731074348834312157, 0,
2.86923266026731074348834312157, 4.20795793242617387384424485764, 5.64369465695127682206552627180, 6.61946123359073797781207247279, 8.292856303885674255776755181762, 9.561810872571921636712154273266, 10.09807116794461824291465021047, 11.83983175074435025619761270369, 12.82561550606998652171882579246
