L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 6·11-s + 2·13-s − 14-s + 16-s − 6·17-s + 19-s + 20-s + 6·22-s − 6·23-s + 25-s − 2·26-s + 28-s + 6·29-s + 8·31-s − 32-s + 6·34-s + 35-s + 2·37-s − 38-s − 40-s − 12·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.80·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s + 0.223·20-s + 1.27·22-s − 1.25·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s − 0.162·38-s − 0.158·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.165455355\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.165455355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−16.17884930457644, −15.88633303806070, −15.47410604386457, −14.89226048042717, −13.86695928188375, −13.59185404006709, −13.14213442719026, −12.14927741311147, −11.86126981168006, −10.90762031293182, −10.49653622122838, −10.20222568485305, −9.373383740769240, −8.586595892453121, −8.323094892862098, −7.610856869714602, −6.975291612275798, −6.130132528856049, −5.721799850653873, −4.789455417042407, −4.246626275596873, −2.944399779974235, −2.473718562980056, −1.684626047350675, −0.5356034188024057,
0.5356034188024057, 1.684626047350675, 2.473718562980056, 2.944399779974235, 4.246626275596873, 4.789455417042407, 5.721799850653873, 6.130132528856049, 6.975291612275798, 7.610856869714602, 8.323094892862098, 8.586595892453121, 9.373383740769240, 10.20222568485305, 10.49653622122838, 10.90762031293182, 11.86126981168006, 12.14927741311147, 13.14213442719026, 13.59185404006709, 13.86695928188375, 14.89226048042717, 15.47410604386457, 15.88633303806070, 16.17884930457644