| L(s) = 1 | − i·2-s − 4-s + 2.16i·5-s − 3.16i·7-s + i·8-s + 2.16·10-s + 5.16i·11-s − 3.16·14-s + 16-s + 3·17-s − 1.16i·19-s − 2.16i·20-s + 5.16·22-s − 0.837·23-s + 0.324·25-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.966i·5-s − 1.19i·7-s + 0.353i·8-s + 0.683·10-s + 1.55i·11-s − 0.845·14-s + 0.250·16-s + 0.727·17-s − 0.266i·19-s − 0.483i·20-s + 1.10·22-s − 0.174·23-s + 0.0649·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233366396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233366396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.16iT - 5T^{2} \) |
| 7 | \( 1 + 3.16iT - 7T^{2} \) |
| 11 | \( 1 - 5.16iT - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 1.16iT - 19T^{2} \) |
| 23 | \( 1 + 0.837T + 23T^{2} \) |
| 29 | \( 1 + 8.16T + 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 + 3iT - 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 - 11.1iT - 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 6.16T + 61T^{2} \) |
| 67 | \( 1 - 9.16iT - 67T^{2} \) |
| 71 | \( 1 - 0.837iT - 71T^{2} \) |
| 73 | \( 1 + iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−9.095577177813343082869473765466, −7.84520911212441590777769156287, −7.38092741177930270009359141922, −6.79730687128966345324469992783, −5.76296167729791885818461369837, −4.64682888394321035655810921003, −4.06410327303661030201922450311, −3.19669583111367030557790752152, −2.27522948665357110017015606699, −1.21199402775712060648801083768,
0.41692679408755152566818021188, 1.73445627843887888560621912941, 3.10715618935577854971358166942, 3.90631696916712906654704285028, 5.16923052781664740994861025789, 5.50910888211699392465290536662, 6.07902603530877703420910726567, 7.11825044254590252642554573712, 8.095834159115059504121811338449, 8.572871632172277478795564565931
