| L(s) = 1 | + 3·2-s + 4-s + 15·5-s − 25·7-s − 21·8-s + 45·10-s − 15·11-s + 20·13-s − 75·14-s − 71·16-s + 72·17-s + 2·19-s + 15·20-s − 45·22-s + 114·23-s + 100·25-s + 60·26-s − 25·28-s + 30·29-s + 101·31-s − 45·32-s + 216·34-s − 375·35-s − 430·37-s + 6·38-s − 315·40-s − 30·41-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s + 1.34·5-s − 1.34·7-s − 0.928·8-s + 1.42·10-s − 0.411·11-s + 0.426·13-s − 1.43·14-s − 1.10·16-s + 1.02·17-s + 0.0241·19-s + 0.167·20-s − 0.436·22-s + 1.03·23-s + 4/5·25-s + 0.452·26-s − 0.168·28-s + 0.192·29-s + 0.585·31-s − 0.248·32-s + 1.08·34-s − 1.81·35-s − 1.91·37-s + 0.0256·38-s − 1.24·40-s − 0.114·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.798681198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798681198\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 5 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 25 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 72 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 T + p^{3} T^{2} \) |
| 23 | \( 1 - 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 101 T + p^{3} T^{2} \) |
| 37 | \( 1 + 430 T + p^{3} T^{2} \) |
| 41 | \( 1 + 30 T + p^{3} T^{2} \) |
| 43 | \( 1 - 110 T + p^{3} T^{2} \) |
| 47 | \( 1 + 330 T + p^{3} T^{2} \) |
| 53 | \( 1 - 621 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 376 T + p^{3} T^{2} \) |
| 67 | \( 1 + 250 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 - 785 T + p^{3} T^{2} \) |
| 79 | \( 1 - 488 T + p^{3} T^{2} \) |
| 83 | \( 1 - 489 T + p^{3} T^{2} \) |
| 89 | \( 1 + 450 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1105 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.73915104352959025967613283304, −15.40179015640148201109537423294, −13.97383315415934176511861245918, −13.29088189313772473035869846756, −12.35947926676341017107510539141, −10.22764950027732158419172506363, −9.140669870214017294457034289509, −6.47382907228091722472211840040, −5.38851828478240733561723784552, −3.15305079314965744138773144565,
3.15305079314965744138773144565, 5.38851828478240733561723784552, 6.47382907228091722472211840040, 9.140669870214017294457034289509, 10.22764950027732158419172506363, 12.35947926676341017107510539141, 13.29088189313772473035869846756, 13.97383315415934176511861245918, 15.40179015640148201109537423294, 16.73915104352959025967613283304
