| L(s) = 1 | − 3i·3-s + 2.82i·5-s + 14.1·7-s − 9·9-s − 20i·11-s − 39.5i·13-s + 8.48·15-s − 34·17-s + 52i·19-s − 42.4i·21-s + 62.2·23-s + 117·25-s + 27i·27-s − 200. i·29-s + 110.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.252i·5-s + 0.763·7-s − 0.333·9-s − 0.548i·11-s − 0.844i·13-s + 0.146·15-s − 0.485·17-s + 0.627i·19-s − 0.440i·21-s + 0.564·23-s + 0.936·25-s + 0.192i·27-s − 1.28i·29-s + 0.639·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.786716837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.786716837\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 - 2.82iT - 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 11 | \( 1 + 20iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 39.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 34T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 271. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 26T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 681. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 364iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 735. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 338T + 3.89e5T^{2} \) |
| 79 | \( 1 - 789.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 234T + 7.04e5T^{2} \) |
| 97 | \( 1 + 178T + 9.12e5T^{2} \) |
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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
−10.90779437708161555587183008371, −9.843666292882086972939124332789, −8.540323970249820788423661850559, −7.991475485402841759188535263164, −6.92835586582445081438932703400, −5.91478345948786207914828955378, −4.88424119895091956135116504667, −3.40875374196671540546513024122, −2.11915038568140354187990086097, −0.64700293773732759806390948661,
1.41718853309754787593893911810, 2.93015530507394179874910201207, 4.54507972618124683788052805180, 4.87793183203197561233041376106, 6.40113102970951909394524857088, 7.40226151026692747736564609205, 8.631381732347455679685464074418, 9.178900918635878238044875792915, 10.29018951978385116896614592549, 11.13748351854466840289004521806
