| L(s) = 1 | − 4.58·2-s + 3·3-s + 12.9·4-s − 12.2·5-s − 13.7·6-s + 15.3·7-s − 22.8·8-s + 9·9-s + 56.3·10-s − 41.7·11-s + 38.9·12-s − 23.4·13-s − 70.3·14-s − 36.8·15-s + 0.870·16-s + 123.·17-s − 41.2·18-s + 106.·19-s − 159.·20-s + 46.0·21-s + 191.·22-s − 207.·23-s − 68.6·24-s + 26.2·25-s + 107.·26-s + 27·27-s + 199.·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.577·3-s + 1.62·4-s − 1.09·5-s − 0.935·6-s + 0.828·7-s − 1.01·8-s + 0.333·9-s + 1.78·10-s − 1.14·11-s + 0.937·12-s − 0.501·13-s − 1.34·14-s − 0.634·15-s + 0.0135·16-s + 1.76·17-s − 0.539·18-s + 1.28·19-s − 1.78·20-s + 0.478·21-s + 1.85·22-s − 1.87·23-s − 0.583·24-s + 0.209·25-s + 0.812·26-s + 0.192·27-s + 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 4.58T + 8T^{2} \) |
| 5 | \( 1 + 12.2T + 125T^{2} \) |
| 7 | \( 1 - 15.3T + 343T^{2} \) |
| 11 | \( 1 + 41.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 207.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 53.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 80.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 5.12T + 1.03e5T^{2} \) |
| 53 | \( 1 + 260.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 35.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 635.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 644.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 531.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 594.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 95.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 964.T + 9.12e5T^{2} \) |
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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
−11.56145825081589440025304004999, −10.40431384428572981715729801812, −9.729110876850934454369251929038, −8.412364842605311150912589788182, −7.74284295532676236411256600617, −7.44892986149998045460506035678, −5.23310383944820812807097873916, −3.44316058969392398437979652052, −1.74897375990281298245142082470, 0,
1.74897375990281298245142082470, 3.44316058969392398437979652052, 5.23310383944820812807097873916, 7.44892986149998045460506035678, 7.74284295532676236411256600617, 8.412364842605311150912589788182, 9.729110876850934454369251929038, 10.40431384428572981715729801812, 11.56145825081589440025304004999
