L(s) = 1 | − 0.414i·2-s + 1.82·4-s − 0.828i·7-s − 1.58i·8-s − 0.585·11-s + i·13-s − 0.343·14-s + 3·16-s + 4.82i·17-s − 3.41·19-s + 0.242i·22-s − 1.41i·23-s + 0.414·26-s − 1.51i·28-s + 5.65·29-s + ⋯ |
L(s) = 1 | − 0.292i·2-s + 0.914·4-s − 0.313i·7-s − 0.560i·8-s − 0.176·11-s + 0.277i·13-s − 0.0917·14-s + 0.750·16-s + 1.17i·17-s − 0.783·19-s + 0.0517i·22-s − 0.294i·23-s + 0.0812·26-s − 0.286i·28-s + 1.05·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.423259411\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.423259411\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 7 | \( 1 + 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 3.07iT - 43T^{2} \) |
| 47 | \( 1 + 0.828iT - 47T^{2} \) |
| 53 | \( 1 + 14.4iT - 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 8.82iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 3.65iT - 97T^{2} \) |
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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
−8.407475896176112544241083768794, −8.153018653967961072669564116045, −7.00700420780196888042983857565, −6.51462792840998949256130489340, −5.85121839306330678239625257878, −4.64924855095409495749761893908, −3.89670635573799835399249257060, −2.88179265295891131507406809131, −2.07301509667445090566488772662, −0.968420804218293394921417045399,
0.982780027961009682797979716689, 2.44863491604934681857326631670, 2.80956839783295989854909371676, 4.15929625441110085577752225073, 5.08137915806345189004731557601, 5.91665573185078177904416329058, 6.49406746472901424792792084085, 7.35264044327888647974973623176, 7.893649330843401018120020352784, 8.716955517945283507056243399530