| L(s) = 1 | − 0.317i·3-s + 2.89·9-s + 3.78i·11-s − 1.89·17-s + 5.97i·19-s − 1.87i·27-s + 1.20·33-s − 6.79·41-s + 8.48i·43-s − 7·49-s + 0.603i·51-s + 1.89·57-s + 14.1i·59-s − 16.3i·67-s − 15.6·73-s + ⋯ |
| L(s) = 1 | − 0.183i·3-s + 0.966·9-s + 1.14i·11-s − 0.460·17-s + 1.37i·19-s − 0.360i·27-s + 0.209·33-s − 1.06·41-s + 1.29i·43-s − 49-s + 0.0845i·51-s + 0.251·57-s + 1.84i·59-s − 1.99i·67-s − 1.83·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494910580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494910580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.317iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 5.97iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 16.3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.0iT - 83T^{2} \) |
| 89 | \( 1 - 4.10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.847267170710445227291051260018, −7.950454481569310954593714858457, −7.39559134378712227372552571282, −6.67506228879209466020487153961, −5.95036282545968668499885879007, −4.83810488360004118419695151736, −4.30792648405243523162388824805, −3.35258774200713434548406027476, −2.09517920993132586701589181508, −1.37463243675774357841767654189,
0.46331887140901201831646089021, 1.74400638007951193498457234961, 2.90896661734940116990221698119, 3.74654879658612356268018537287, 4.63587454720973309898370668866, 5.32111319638896904877513300947, 6.33815193125964511761050579022, 6.93476715392897283382717467565, 7.69897615601872846776938692454, 8.675781141188546485463898406131
