| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s − 13-s + 14-s + 16-s + 4·17-s − 3·19-s − 20-s + 22-s − 6·23-s + 25-s − 26-s + 28-s + 4·29-s − 7·31-s + 32-s + 4·34-s − 35-s − 8·37-s − 3·38-s − 40-s − 6·41-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 + 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.688·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 0.742·29-s − 1.25·31-s + 0.176·32-s + 0.685·34-s − 0.169·35-s − 1.31·37-s − 0.486·38-s − 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−16.32699756603787, −16.00395466237999, −15.36361040765546, −14.67733869251370, −14.40885145871050, −13.86395212644873, −13.14679699978338, −12.47000781486791, −12.10470116047655, −11.58249781687807, −10.97417058094337, −10.24063650497108, −9.873923704021851, −8.856953728933922, −8.281239766899605, −7.762431453534324, −6.969545335629342, −6.548446594341733, −5.538722481045025, −5.243920767636238, −4.271856109886674, −3.828503210047173, −3.088555246055485, −2.132735126692690, −1.382963795253304, 0,
1.382963795253304, 2.132735126692690, 3.088555246055485, 3.828503210047173, 4.271856109886674, 5.243920767636238, 5.538722481045025, 6.548446594341733, 6.969545335629342, 7.762431453534324, 8.281239766899605, 8.856953728933922, 9.873923704021851, 10.24063650497108, 10.97417058094337, 11.58249781687807, 12.10470116047655, 12.47000781486791, 13.14679699978338, 13.86395212644873, 14.40885145871050, 14.67733869251370, 15.36361040765546, 16.00395466237999, 16.32699756603787
