| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·7-s − 2·10-s + 3·11-s + 6·13-s + 4·14-s − 4·16-s − 6·17-s + 2·20-s − 6·22-s + 8·23-s + 25-s − 12·26-s − 4·28-s − 7·29-s + 9·31-s + 8·32-s + 12·34-s − 2·35-s − 2·37-s − 6·41-s + 10·43-s + 6·44-s − 16·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s − 0.632·10-s + 0.904·11-s + 1.66·13-s + 1.06·14-s − 16-s − 1.45·17-s + 0.447·20-s − 1.27·22-s + 1.66·23-s + 1/5·25-s − 2.35·26-s − 0.755·28-s − 1.29·29-s + 1.61·31-s + 1.41·32-s + 2.05·34-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 1.52·43-s + 0.904·44-s − 2.35·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−16.34282933144755, −15.78559287572106, −15.45834936958880, −14.69836767548134, −13.79811183769072, −13.44391658520617, −13.04929326117170, −12.25834003626368, −11.32007050572292, −11.02412150903052, −10.65074962290063, −9.711776686360789, −9.365027847329194, −8.899155653931670, −8.502414365771857, −7.735372608527110, −6.867539628814433, −6.502792736488725, −6.113504249696727, −4.985152346187800, −4.244078580201647, −3.440346219583338, −2.641239497506407, −1.607860427970451, −1.123835423275056, 0,
1.123835423275056, 1.607860427970451, 2.641239497506407, 3.440346219583338, 4.244078580201647, 4.985152346187800, 6.113504249696727, 6.502792736488725, 6.867539628814433, 7.735372608527110, 8.502414365771857, 8.899155653931670, 9.365027847329194, 9.711776686360789, 10.65074962290063, 11.02412150903052, 11.32007050572292, 12.25834003626368, 13.04929326117170, 13.44391658520617, 13.79811183769072, 14.69836767548134, 15.45834936958880, 15.78559287572106, 16.34282933144755
