| L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s + 4·7-s + 3·8-s + 9-s − 2·10-s + 11-s + 12-s + 2·13-s − 4·14-s − 2·15-s − 16-s + 2·17-s − 18-s − 2·20-s − 4·21-s − 22-s − 8·23-s − 3·24-s − 25-s − 2·26-s − 27-s − 4·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.447·20-s − 0.872·21-s − 0.213·22-s − 1.66·23-s − 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.965788784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965788784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.53060514881058, −13.94639273729022, −13.73718587101267, −13.35130233028368, −12.39233626562793, −11.89779720155154, −11.62754490733860, −10.72695475109137, −10.43506816506446, −10.01877565492817, −9.464025387424432, −8.780252403867520, −8.329791252110879, −7.900757909710065, −7.362092238876390, −6.493126544937003, −5.942515751427286, −5.464597112607142, −4.799412988359086, −4.340317054168814, −3.776147357218482, −2.551976879788703, −1.789081057796837, −1.322973317644450, −0.6559073000195776,
0.6559073000195776, 1.322973317644450, 1.789081057796837, 2.551976879788703, 3.776147357218482, 4.340317054168814, 4.799412988359086, 5.464597112607142, 5.942515751427286, 6.493126544937003, 7.362092238876390, 7.900757909710065, 8.329791252110879, 8.780252403867520, 9.464025387424432, 10.01877565492817, 10.43506816506446, 10.72695475109137, 11.62754490733860, 11.89779720155154, 12.39233626562793, 13.35130233028368, 13.73718587101267, 13.94639273729022, 14.53060514881058
