| L(s) = 1 | − 2·3-s + 9-s + 11-s − 5·13-s − 6·17-s + 4·19-s + 6·23-s + 4·27-s + 6·29-s − 5·31-s − 2·33-s + 4·37-s + 10·39-s + 11·43-s + 12·51-s − 6·53-s − 8·57-s + 3·59-s − 10·61-s + 8·67-s − 12·69-s − 3·71-s + 13·73-s − 8·79-s − 11·81-s + 3·83-s − 12·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.301·11-s − 1.38·13-s − 1.45·17-s + 0.917·19-s + 1.25·23-s + 0.769·27-s + 1.11·29-s − 0.898·31-s − 0.348·33-s + 0.657·37-s + 1.60·39-s + 1.67·43-s + 1.68·51-s − 0.824·53-s − 1.05·57-s + 0.390·59-s − 1.28·61-s + 0.977·67-s − 1.44·69-s − 0.356·71-s + 1.52·73-s − 0.900·79-s − 1.22·81-s + 0.329·83-s − 1.28·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−12.87850873800150, −12.71464266979723, −12.35397291901971, −11.68869311543453, −11.30486725003602, −11.12351657158781, −10.48846110683669, −10.07033510681853, −9.430184982845570, −9.093744881866649, −8.655424218409668, −7.840276722408008, −7.412166043889453, −6.904766979085243, −6.555524206445697, −5.985043779761543, −5.471838168179554, −4.866200178300695, −4.712492674809107, −4.087945144496135, −3.243410649379807, −2.668954882073290, −2.210103014416144, −1.264484560279482, −0.6887714077042894, 0,
0.6887714077042894, 1.264484560279482, 2.210103014416144, 2.668954882073290, 3.243410649379807, 4.087945144496135, 4.712492674809107, 4.866200178300695, 5.471838168179554, 5.985043779761543, 6.555524206445697, 6.904766979085243, 7.412166043889453, 7.840276722408008, 8.655424218409668, 9.093744881866649, 9.430184982845570, 10.07033510681853, 10.48846110683669, 11.12351657158781, 11.30486725003602, 11.68869311543453, 12.35397291901971, 12.71464266979723, 12.87850873800150
