| L(s) = 1 | − 7-s + 2·17-s + 4·19-s − 4·23-s − 5·25-s − 6·29-s − 4·31-s + 6·37-s + 6·41-s + 4·43-s − 12·47-s + 49-s − 4·53-s + 8·61-s − 12·67-s + 4·71-s − 4·73-s + 4·83-s − 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 25-s − 1.11·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.549·53-s + 1.02·61-s − 1.46·67-s + 0.474·71-s − 0.468·73-s + 0.439·83-s − 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.80415387782848, −13.25684115364678, −12.83038362706387, −12.43104923661665, −11.79851791261379, −11.34979508219131, −11.07791218996053, −10.21047463026777, −9.917272483346990, −9.378550279850462, −9.137074777704393, −8.186295850066544, −7.939574198489908, −7.343074799557060, −6.972937976327379, −6.048995451538175, −5.882451656769037, −5.356460214208040, −4.564688408979182, −4.076383891246439, −3.427892097241263, −3.037281015261056, −2.163291541341052, −1.672047751292533, −0.8003845535159318, 0,
0.8003845535159318, 1.672047751292533, 2.163291541341052, 3.037281015261056, 3.427892097241263, 4.076383891246439, 4.564688408979182, 5.356460214208040, 5.882451656769037, 6.048995451538175, 6.972937976327379, 7.343074799557060, 7.939574198489908, 8.186295850066544, 9.137074777704393, 9.378550279850462, 9.917272483346990, 10.21047463026777, 11.07791218996053, 11.34979508219131, 11.79851791261379, 12.43104923661665, 12.83038362706387, 13.25684115364678, 13.80415387782848
