| L(s) = 1 | − 3-s − 5-s − 0.855i·7-s + 9-s − 2.02i·11-s − 6.27i·13-s + 15-s + 4.01·17-s + (−0.531 − 4.32i)19-s + 0.855i·21-s + 2.66i·23-s + 25-s − 27-s − 2.52i·29-s + 2.57·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.323i·7-s + 0.333·9-s − 0.612i·11-s − 1.73i·13-s + 0.258·15-s + 0.972·17-s + (−0.121 − 0.992i)19-s + 0.186i·21-s + 0.554i·23-s + 0.200·25-s − 0.192·27-s − 0.469i·29-s + 0.463·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9627380496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9627380496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + (0.531 + 4.32i)T \) |
| good | 7 | \( 1 + 0.855iT - 7T^{2} \) |
| 11 | \( 1 + 2.02iT - 11T^{2} \) |
| 13 | \( 1 + 6.27iT - 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 23 | \( 1 - 2.66iT - 23T^{2} \) |
| 29 | \( 1 + 2.52iT - 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + 1.25iT - 37T^{2} \) |
| 41 | \( 1 - 0.709iT - 41T^{2} \) |
| 43 | \( 1 - 2.20iT - 43T^{2} \) |
| 47 | \( 1 + 1.98iT - 47T^{2} \) |
| 53 | \( 1 + 3.05iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.23T + 61T^{2} \) |
| 67 | \( 1 - 4.14T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 + 9.52T + 73T^{2} \) |
| 79 | \( 1 - 4.17T + 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 5.31iT - 97T^{2} \) |
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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
−7.924391448374307401493308765923, −7.43126505977776570485117657869, −6.56537119240822441966834121090, −5.69703630819075047539593192284, −5.26027408533818742547852972216, −4.30235304525876035339985767184, −3.41431885616697792275153124134, −2.73808021631902845641273809908, −1.13281320117927306296765050782, −0.34403977689878826506364205103,
1.27738273886514017098247585403, 2.17385765622065146133680318748, 3.39395518310767996282422493786, 4.27515152082467394209206683997, 4.78839649749534193017353491434, 5.78210097601192985435271903687, 6.40703789537766441781055946779, 7.17162768576941685310490220456, 7.74473834974097713730924818765, 8.678842683837339353314456485088
