| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s − 2·7-s + 9-s − 2·10-s + 2·11-s + 2·12-s − 13-s + 4·14-s + 15-s − 4·16-s + 3·17-s − 2·18-s + 5·19-s + 2·20-s − 2·21-s − 4·22-s + 4·23-s − 4·25-s + 2·26-s + 27-s − 4·28-s + 5·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 0.755·7-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.577·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.727·17-s − 0.471·18-s + 1.14·19-s + 0.447·20-s − 0.436·21-s − 0.852·22-s + 0.834·23-s − 4/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.928·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 537 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 537 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9149453527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9149453527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 179 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.38936623791296325354512398373, −9.532768221874399136814290801332, −9.401125101518721919206423907363, −8.321281048624554700373284766719, −7.42702072574904163820367463689, −6.70290380883587671865344475874, −5.40518731409452841662999014454, −3.80793274389673457721829277471, −2.52125713735440015843855024278, −1.10075550837814949351322518252,
1.10075550837814949351322518252, 2.52125713735440015843855024278, 3.80793274389673457721829277471, 5.40518731409452841662999014454, 6.70290380883587671865344475874, 7.42702072574904163820367463689, 8.321281048624554700373284766719, 9.401125101518721919206423907363, 9.532768221874399136814290801332, 10.38936623791296325354512398373
