| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 2·11-s + 4·13-s − 15-s − 17-s + 2·19-s − 21-s + 4·23-s + 25-s − 27-s − 4·29-s + 8·31-s + 2·33-s + 35-s − 8·37-s − 4·39-s + 10·41-s + 45-s + 12·47-s + 49-s + 51-s − 6·53-s − 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s − 0.242·17-s + 0.458·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.43·31-s + 0.348·33-s + 0.169·35-s − 1.31·37-s − 0.640·39-s + 1.56·41-s + 0.149·45-s + 1.75·47-s + 1/7·49-s + 0.140·51-s − 0.824·53-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.52844313431904, −14.01005493979331, −13.61300921041664, −13.09974824481203, −12.64700001056720, −12.05831530083278, −11.51225034883944, −10.91078855756707, −10.72516543209576, −10.12999595839774, −9.460854775913335, −8.958683851418637, −8.455012198949890, −7.792445163048790, −7.262059135906793, −6.694244626947066, −6.017157191097945, −5.640261140906863, −5.108192619668005, −4.429808620218209, −3.912490795900285, −2.995041880465135, −2.512958438545040, −1.480604607566778, −1.098761681714976, 0,
1.098761681714976, 1.480604607566778, 2.512958438545040, 2.995041880465135, 3.912490795900285, 4.429808620218209, 5.108192619668005, 5.640261140906863, 6.017157191097945, 6.694244626947066, 7.262059135906793, 7.792445163048790, 8.455012198949890, 8.958683851418637, 9.460854775913335, 10.12999595839774, 10.72516543209576, 10.91078855756707, 11.51225034883944, 12.05831530083278, 12.64700001056720, 13.09974824481203, 13.61300921041664, 14.01005493979331, 14.52844313431904
