| L(s) = 1 | − i·3-s + (−0.676 − 2.13i)5-s − 4.23i·7-s − 9-s − 1.39·11-s − 6.36i·13-s + (−2.13 + 0.676i)15-s + 4.10i·17-s − 5.15·19-s − 4.23·21-s + 1.91i·23-s + (−4.08 + 2.88i)25-s + i·27-s − 5.40·29-s + 0.479·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.302 − 0.953i)5-s − 1.59i·7-s − 0.333·9-s − 0.419·11-s − 1.76i·13-s + (−0.550 + 0.174i)15-s + 0.996i·17-s − 1.18·19-s − 0.923·21-s + 0.398i·23-s + (−0.817 + 0.576i)25-s + 0.192i·27-s − 1.00·29-s + 0.0861·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6850146004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6850146004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.676 + 2.13i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 + 4.23iT - 7T^{2} \) |
| 11 | \( 1 + 1.39T + 11T^{2} \) |
| 13 | \( 1 + 6.36iT - 13T^{2} \) |
| 17 | \( 1 - 4.10iT - 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 - 1.91iT - 23T^{2} \) |
| 29 | \( 1 + 5.40T + 29T^{2} \) |
| 31 | \( 1 - 0.479T + 31T^{2} \) |
| 37 | \( 1 + 2.52iT - 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 + 0.604iT - 43T^{2} \) |
| 47 | \( 1 - 0.216iT - 47T^{2} \) |
| 53 | \( 1 + 2.89iT - 53T^{2} \) |
| 59 | \( 1 + 0.893T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 71 | \( 1 - 8.26T + 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 12.0iT - 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 0.636iT - 97T^{2} \) |
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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
−7.87175219874757321654971218108, −7.49710289324735057304572925127, −6.49227024005021930920995650446, −5.68823695592223549149522735241, −4.94250427622602336897962697717, −3.99544185883370254133184932280, −3.45619563262720596305804715360, −2.06285640158895099132425695162, −0.987612001612939003008484699621, −0.22020202201143359717332049228,
2.16326571084433826558601202383, 2.50249777864993982852943016070, 3.57284375708984348270721504873, 4.43675953881673857922128197895, 5.18605310435184457507133699477, 6.12708749080061602199225754246, 6.59442548962907838088651766299, 7.48474075690998466287218617682, 8.370792957372999048094807266157, 9.079233494331647464647855685564
