| L(s) = 1 | − 3-s − 5-s + 9-s − 3·11-s − 4·13-s + 15-s − 4·17-s + 19-s + 4·23-s − 4·25-s − 27-s − 7·29-s − 7·31-s + 3·33-s − 4·37-s + 4·39-s + 12·41-s + 6·43-s − 45-s − 10·47-s + 4·51-s + 13·53-s + 3·55-s − 57-s + 59-s + 14·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.258·15-s − 0.970·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s − 0.192·27-s − 1.29·29-s − 1.25·31-s + 0.522·33-s − 0.657·37-s + 0.640·39-s + 1.87·41-s + 0.914·43-s − 0.149·45-s − 1.45·47-s + 0.560·51-s + 1.78·53-s + 0.404·55-s − 0.132·57-s + 0.130·59-s + 1.79·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
| 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
−14.95096746167604, −14.51481158596586, −13.88458299388782, −13.06666663490162, −12.87710449427742, −12.47846055830113, −11.60663499125229, −11.28501092313081, −10.98225189464835, −10.11557109721349, −9.886545471403457, −9.055242991187375, −8.713898905340937, −7.784973261967691, −7.331490609307869, −7.158797339597449, −6.254013431335778, −5.502708039794791, −5.285541857279048, −4.493237413706474, −4.000327359461233, −3.237980117562455, −2.375695946151310, −1.933739710343390, −0.6863281222913220, 0,
0.6863281222913220, 1.933739710343390, 2.375695946151310, 3.237980117562455, 4.000327359461233, 4.493237413706474, 5.285541857279048, 5.502708039794791, 6.254013431335778, 7.158797339597449, 7.331490609307869, 7.784973261967691, 8.713898905340937, 9.055242991187375, 9.886545471403457, 10.11557109721349, 10.98225189464835, 11.28501092313081, 11.60663499125229, 12.47846055830113, 12.87710449427742, 13.06666663490162, 13.88458299388782, 14.51481158596586, 14.95096746167604
