| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−14.63916320984897, −13.96866608687381, −13.57440060796830, −13.09255845132126, −12.53700719251514, −11.90820643595722, −11.57205999193530, −11.06704954823045, −10.42133529830218, −9.915397984258024, −9.604585531806914, −8.725417054004016, −8.167242687759772, −8.138679059036627, −7.217025789334415, −6.537950264556073, −6.306031732559428, −5.531117263327693, −4.882894653912617, −4.387969481402981, −3.825651143297415, −3.104843895174687, −2.259820255763959, −1.884626462494536, −0.9055025743709319, 0,
0.9055025743709319, 1.884626462494536, 2.259820255763959, 3.104843895174687, 3.825651143297415, 4.387969481402981, 4.882894653912617, 5.531117263327693, 6.306031732559428, 6.537950264556073, 7.217025789334415, 8.138679059036627, 8.167242687759772, 8.725417054004016, 9.604585531806914, 9.915397984258024, 10.42133529830218, 11.06704954823045, 11.57205999193530, 11.90820643595722, 12.53700719251514, 13.09255845132126, 13.57440060796830, 13.96866608687381, 14.63916320984897
