| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 2·7-s + 3·8-s + 9-s − 10-s − 11-s − 12-s − 5·13-s + 2·14-s + 15-s − 16-s + 4·17-s − 18-s − 4·19-s − 20-s − 2·21-s + 22-s − 23-s + 3·24-s + 25-s + 5·26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 1.38·13-s + 0.534·14-s + 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s − 0.208·23-s + 0.612·24-s + 1/5·25-s + 0.980·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.87486617746256, −14.43388028418761, −14.10386904560765, −13.23428542887753, −12.95234341532903, −12.71180641805515, −11.91376915640712, −11.19435331271648, −10.49270652496567, −9.992002620269496, −9.686278184615816, −9.348802946810189, −8.719003130974272, −8.001512891943177, −7.747922821443720, −7.106419182577278, −6.444122399541344, −5.790911489206227, −4.941281913524233, −4.702466457433741, −3.671974297118515, −3.294699554229905, −2.284734747960859, −1.920791982674916, −0.8119650634924072, 0,
0.8119650634924072, 1.920791982674916, 2.284734747960859, 3.294699554229905, 3.671974297118515, 4.702466457433741, 4.941281913524233, 5.790911489206227, 6.444122399541344, 7.106419182577278, 7.747922821443720, 8.001512891943177, 8.719003130974272, 9.348802946810189, 9.686278184615816, 9.992002620269496, 10.49270652496567, 11.19435331271648, 11.91376915640712, 12.71180641805515, 12.95234341532903, 13.23428542887753, 14.10386904560765, 14.43388028418761, 14.87486617746256
