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 | \( 1 + 3.11e3T + 8.85e7T^{2} \) |
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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
−10.03140458265505713633628037089, −8.966249655666219795589435064595, −8.337046453597408411003888483127, −7.56282976026679032980354310670, −6.33467772669143964382928789587, −5.51979990959826472319262826217, −4.30059573518722625821302792683, −3.67680750873840046711878126382, −2.56146021108641373418885146370, −0.807899840943055689374903361463,
0.37236780708431007065446656595, 1.62931822960314352190964424557, 2.91364359985864332371248168753, 4.04406195369995341381682360959, 4.90757140812339736197171641745, 6.17278846313285103581711043190, 7.14557873973935352004009149264, 7.62466482077843759839955294838, 8.580038302570676420503218984955, 9.398008940827234403630167392764