| L(s) = 1 | + 1.80·2-s − 12.7·4-s − 12.1i·7-s − 51.7·8-s − 60.4i·11-s + 168. i·13-s − 21.8i·14-s + 110.·16-s + 377.·17-s + 45.1·19-s − 108. i·22-s + 1.37·23-s + 304. i·26-s + 154. i·28-s + 1.40e3i·29-s + ⋯ |
| L(s) = 1 | + 0.450·2-s − 0.797·4-s − 0.247i·7-s − 0.809·8-s − 0.499i·11-s + 0.999i·13-s − 0.111i·14-s + 0.432·16-s + 1.30·17-s + 0.125·19-s − 0.224i·22-s + 0.00259·23-s + 0.450i·26-s + 0.197i·28-s + 1.66i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.804707263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804707263\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.80T + 16T^{2} \) |
| 7 | \( 1 + 12.1iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 60.4iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 168. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 377.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 45.1T + 1.30e5T^{2} \) |
| 23 | \( 1 - 1.37T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.40e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.60e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.32e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.35e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 468. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 3.91e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.22e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.54e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 254.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.49e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 4.14e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.71e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 80.2T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.15e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 5.87e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 8.92e3iT - 8.85e7T^{2} \) |
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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
−11.91231182318655486134311555811, −10.69135218323764058281949229411, −9.621587615017458377395023579416, −8.799843247938276758551853150651, −7.71536813360458066582672438160, −6.36598658556682671689449891413, −5.27467415921823686286481364890, −4.20511691295616739139125254928, −3.09586017910965287856547132118, −1.02248995727523575244219373271,
0.72745761274216349119332010268, 2.79812597244491223136346645496, 4.04617123064304351320820822272, 5.20674929831187254774854504754, 6.04885607223451994648258291801, 7.63951498895185098537872470114, 8.494426597691109042346218795348, 9.689252894718076948436159990640, 10.31604711862415101338927379176, 11.90447867688768258281458849845
