| 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 | \( 1 - 1.63e3iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47177426064802263608151346761, −9.347945959485985767297050085268, −8.938881734441480588555998880665, −7.913352370001075159685331044057, −6.52540282771831746062670303778, −6.08985261807926313753308484821, −4.53397163324778011718774849468, −4.00563703197745381294656309836, −2.62445535050150252813210834323, −1.11661554429713011183707908193,
0.77255853202351811059780156652, 1.88234489221942233959947996107, 3.38064279740927582896832935625, 4.36417776808573105888307903932, 5.69668832742928822351224076743, 6.64125801374809805071914994705, 7.25289629556677057149254609618, 8.402252492826185807553042885624, 9.171320395518647947085934709538, 10.00740828160630579661965223252
