| L(s) = 1 | − 5.19i·3-s − 23.7·5-s − 9.38i·7-s − 27·9-s + 112. i·11-s + 56.5·13-s + 123. i·15-s − 79.9·17-s − 211. i·19-s − 48.7·21-s + 217. i·23-s − 60.9·25-s + 140. i·27-s + 616.·29-s + 1.11e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.949·5-s − 0.191i·7-s − 0.333·9-s + 0.925i·11-s + 0.334·13-s + 0.548i·15-s − 0.276·17-s − 0.585i·19-s − 0.110·21-s + 0.410i·23-s − 0.0975·25-s + 0.192i·27-s + 0.733·29-s + 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.164778197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164778197\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| good | 5 | \( 1 + 23.7T + 625T^{2} \) |
| 7 | \( 1 + 9.38iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 112. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 56.5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 79.9T + 8.35e4T^{2} \) |
| 19 | \( 1 + 211. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 217. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 616.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.11e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 802.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.41e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.13e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 3.59e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 833.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.30e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.78e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.02e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 9.48e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 266.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.75e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 7.28e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.41e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 3.11e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.398008940827234403630167392764, −8.580038302570676420503218984955, −7.62466482077843759839955294838, −7.14557873973935352004009149264, −6.17278846313285103581711043190, −4.90757140812339736197171641745, −4.04406195369995341381682360959, −2.91364359985864332371248168753, −1.62931822960314352190964424557, −0.37236780708431007065446656595,
0.807899840943055689374903361463, 2.56146021108641373418885146370, 3.67680750873840046711878126382, 4.30059573518722625821302792683, 5.51979990959826472319262826217, 6.33467772669143964382928789587, 7.56282976026679032980354310670, 8.337046453597408411003888483127, 8.966249655666219795589435064595, 10.03140458265505713633628037089
