| L(s) = 1 | − 3i·3-s − 4i·7-s − 9·9-s + 72·11-s − 6i·13-s − 38i·17-s − 52·19-s − 12·21-s + 152i·23-s + 27i·27-s + 78·29-s + 120·31-s − 216i·33-s + 150i·37-s − 18·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.215i·7-s − 0.333·9-s + 1.97·11-s − 0.128i·13-s − 0.542i·17-s − 0.627·19-s − 0.124·21-s + 1.37i·23-s + 0.192i·27-s + 0.499·29-s + 0.695·31-s − 1.13i·33-s + 0.666i·37-s − 0.0739·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.148052348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148052348\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 343T^{2} \) |
| 11 | \( 1 - 72T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 38iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 52T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 78T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 362T + 6.89e4T^{2} \) |
| 43 | \( 1 + 484iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 280iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 670iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 696T + 2.05e5T^{2} \) |
| 61 | \( 1 - 222T + 2.26e5T^{2} \) |
| 67 | \( 1 - 4iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 96T + 3.57e5T^{2} \) |
| 73 | \( 1 - 178iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 632T + 4.93e5T^{2} \) |
| 83 | \( 1 + 612iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 994T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.63e3iT - 9.12e5T^{2} \) |
| 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
−10.00740828160630579661965223252, −9.171320395518647947085934709538, −8.402252492826185807553042885624, −7.25289629556677057149254609618, −6.64125801374809805071914994705, −5.69668832742928822351224076743, −4.36417776808573105888307903932, −3.38064279740927582896832935625, −1.88234489221942233959947996107, −0.77255853202351811059780156652,
1.11661554429713011183707908193, 2.62445535050150252813210834323, 4.00563703197745381294656309836, 4.53397163324778011718774849468, 6.08985261807926313753308484821, 6.52540282771831746062670303778, 7.913352370001075159685331044057, 8.938881734441480588555998880665, 9.347945959485985767297050085268, 10.47177426064802263608151346761
