| L(s) = 1 | − 1.61i·2-s − i·3-s − 0.618·4-s − 1.61·6-s − 2.23i·7-s − 2.23i·8-s − 9-s − 4·11-s + 0.618i·12-s − 2.47i·13-s − 3.61·14-s − 4.85·16-s + 3.23i·17-s + 1.61i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 1.14i·2-s − 0.577i·3-s − 0.309·4-s − 0.660·6-s − 0.845i·7-s − 0.790i·8-s − 0.333·9-s − 1.20·11-s + 0.178i·12-s − 0.685i·13-s − 0.966·14-s − 1.21·16-s + 0.784i·17-s + 0.381i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.048987314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048987314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61iT - 2T^{2} \) |
| 7 | \( 1 + 2.23iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2.47iT - 13T^{2} \) |
| 17 | \( 1 - 3.23iT - 17T^{2} \) |
| 23 | \( 1 + 1.23iT - 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 7.23iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8.47iT - 47T^{2} \) |
| 53 | \( 1 + 5iT - 53T^{2} \) |
| 59 | \( 1 - 1.29T + 59T^{2} \) |
| 61 | \( 1 + 9.94T + 61T^{2} \) |
| 67 | \( 1 + 6.94iT - 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 5.47iT - 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 10.9iT - 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 11.7iT - 97T^{2} \) |
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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
−9.203416628470339016627709094606, −8.064069159776848731181899226965, −7.52760199730064284716334605273, −6.62694170130509033553134261255, −5.65670472493239329103341032846, −4.51454906045621808413549782053, −3.46772875231093882600242374618, −2.66696231278990812457603910566, −1.59038933078767992311029568140, −0.39610447143725474632307808322,
2.18742098784607666408074281651, 3.10369145586348187922038542723, 4.61311350784467770169161754741, 5.26532051184216963481459456202, 5.88420346710924251514826918127, 6.86651164417588873683717278384, 7.60100615752859259962530268432, 8.475935031902701953149010554117, 9.003861157464723395885610922227, 9.930712048542356054513869308304
