| L(s) = 1 | − 5.19·3-s − 30.1i·5-s + 52.3i·7-s + 27·9-s + 90.0·11-s − 60.3i·13-s + 156. i·15-s − 338·17-s + 6.92·19-s − 271. i·21-s + 732. i·23-s − 287·25-s − 140.·27-s + 1.29e3i·29-s − 1.30e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.20i·5-s + 1.06i·7-s + 0.333·9-s + 0.744·11-s − 0.357i·13-s + 0.697i·15-s − 1.16·17-s + 0.0191·19-s − 0.616i·21-s + 1.38i·23-s − 0.459·25-s − 0.192·27-s + 1.54i·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.537972458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537972458\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19T \) |
| good | 5 | \( 1 + 30.1iT - 625T^{2} \) |
| 7 | \( 1 - 52.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 90.0T + 1.46e4T^{2} \) |
| 13 | \( 1 + 60.3iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 338T + 8.35e4T^{2} \) |
| 19 | \( 1 - 6.92T + 1.30e5T^{2} \) |
| 23 | \( 1 - 732. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.29e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.30e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 241. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 578T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.02e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.19e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.44e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.19e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 6.40e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 8.26e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.28e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.12e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.31e4T + 4.74e7T^{2} \) |
| 89 | \( 1 - 910T + 6.27e7T^{2} \) |
| 97 | \( 1 - 5.42e3T + 8.85e7T^{2} \) |
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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
−10.95741564748408011774224125615, −9.458330236891885032563300988918, −9.072781494533863324990045920795, −8.089400363117075459639149619343, −6.77871611233654619397732711077, −5.67879865361347589743628666262, −5.04439148759725351036286504265, −3.86828312726056042266193906857, −2.09126214270688890135136915585, −0.817161908041554561167793361454,
0.69410615792173018095000668515, 2.34428443615831942889709409196, 3.79538142655755248563776694299, 4.63183664291523399896086026176, 6.37140804535537813900463707570, 6.67771946087191592682582015915, 7.62974961731876410226981001825, 8.976074347651282744798686923629, 10.12691540007233031257325829188, 10.77103797110207910071592426362
