| L(s) = 1 | − 9·9-s + 144·11-s − 104·19-s + 156·29-s + 240·31-s + 724·41-s + 670·49-s − 1.39e3·59-s + 444·61-s + 192·71-s + 1.26e3·79-s + 81·81-s − 1.98e3·89-s − 1.29e3·99-s + 1.78e3·101-s − 892·109-s + 1.28e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 3.94·11-s − 1.25·19-s + 0.998·29-s + 1.39·31-s + 2.75·41-s + 1.95·49-s − 3.07·59-s + 0.931·61-s + 0.320·71-s + 1.80·79-s + 1/9·81-s − 2.36·89-s − 1.31·99-s + 1.75·101-s − 0.783·109-s + 9.68·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.98·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.614128893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.614128893\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4358 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8382 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1230 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 78806 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 362 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 75242 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 129246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 151146 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 696 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 222 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 601510 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 746350 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 p T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 769030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 994 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 844610 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47177426064802263608151346761, −10.00740828160630579661965223252, −9.347945959485985767297050085268, −9.171320395518647947085934709538, −8.938881734441480588555998880665, −8.402252492826185807553042885624, −7.913352370001075159685331044057, −7.25289629556677057149254609618, −6.64125801374809805071914994705, −6.52540282771831746062670303778, −6.08985261807926313753308484821, −5.69668832742928822351224076743, −4.53397163324778011718774849468, −4.36417776808573105888307903932, −4.00563703197745381294656309836, −3.38064279740927582896832935625, −2.62445535050150252813210834323, −1.88234489221942233959947996107, −1.11661554429713011183707908193, −0.77255853202351811059780156652,
0.77255853202351811059780156652, 1.11661554429713011183707908193, 1.88234489221942233959947996107, 2.62445535050150252813210834323, 3.38064279740927582896832935625, 4.00563703197745381294656309836, 4.36417776808573105888307903932, 4.53397163324778011718774849468, 5.69668832742928822351224076743, 6.08985261807926313753308484821, 6.52540282771831746062670303778, 6.64125801374809805071914994705, 7.25289629556677057149254609618, 7.913352370001075159685331044057, 8.402252492826185807553042885624, 8.938881734441480588555998880665, 9.171320395518647947085934709538, 9.347945959485985767297050085268, 10.00740828160630579661965223252, 10.47177426064802263608151346761
