| L(s) = 1 | + 4·7-s − 3·11-s + 2·13-s − 6·17-s − 5·19-s + 8·23-s + 3·29-s − 5·31-s + 4·37-s − 9·41-s − 4·43-s − 2·47-s + 9·49-s + 3·59-s + 6·61-s + 6·67-s + 5·71-s − 12·77-s − 8·79-s + 6·83-s − 89-s + 8·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.904·11-s + 0.554·13-s − 1.45·17-s − 1.14·19-s + 1.66·23-s + 0.557·29-s − 0.898·31-s + 0.657·37-s − 1.40·41-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.390·59-s + 0.768·61-s + 0.733·67-s + 0.593·71-s − 1.36·77-s − 0.900·79-s + 0.658·83-s − 0.105·89-s + 0.838·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.62941757212447, −13.96169231479799, −13.36006944183295, −13.05854295475268, −12.64249815691916, −11.77134332053173, −11.32606562109564, −10.96082030013222, −10.64651176786219, −10.04150446694792, −9.176734841943468, −8.743756765019292, −8.263519269360284, −8.013781180513759, −7.066243726300395, −6.833674400280805, −6.098737350375963, −5.276081293551587, −4.979549523016532, −4.461890372449145, −3.837914432512838, −2.986975914610972, −2.266340578320396, −1.815477301754546, −0.9939884782721783, 0,
0.9939884782721783, 1.815477301754546, 2.266340578320396, 2.986975914610972, 3.837914432512838, 4.461890372449145, 4.979549523016532, 5.276081293551587, 6.098737350375963, 6.833674400280805, 7.066243726300395, 8.013781180513759, 8.263519269360284, 8.743756765019292, 9.176734841943468, 10.04150446694792, 10.64651176786219, 10.96082030013222, 11.32606562109564, 11.77134332053173, 12.64249815691916, 13.05854295475268, 13.36006944183295, 13.96169231479799, 14.62941757212447
