L(s) = 1 | + (−1.41 − 0.0591i)2-s − 3-s + (1.99 + 0.167i)4-s + (1.41 + 0.0591i)6-s − 1.33i·7-s + (−2.80 − 0.353i)8-s + 9-s + 2.94i·11-s + (−1.99 − 0.167i)12-s + 2.04·13-s + (−0.0788 + 1.88i)14-s + (3.94 + 0.665i)16-s − 3.61i·17-s + (−1.41 − 0.0591i)18-s − 5.35i·19-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0418i)2-s − 0.577·3-s + (0.996 + 0.0835i)4-s + (0.576 + 0.0241i)6-s − 0.504i·7-s + (−0.992 − 0.125i)8-s + 0.333·9-s + 0.887i·11-s + (−0.575 − 0.0482i)12-s + 0.566·13-s + (−0.0210 + 0.503i)14-s + (0.986 + 0.166i)16-s − 0.876i·17-s + (−0.333 − 0.0139i)18-s − 1.22i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.731645 - 0.221956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.731645 - 0.221956i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0591i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.33iT - 7T^{2} \) |
| 11 | \( 1 - 2.94iT - 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 3.61iT - 17T^{2} \) |
| 19 | \( 1 + 5.35iT - 19T^{2} \) |
| 23 | \( 1 - 8.59iT - 23T^{2} \) |
| 29 | \( 1 + 5.26iT - 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 - 6.55T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 + 9.97iT - 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 + 4.75iT - 59T^{2} \) |
| 61 | \( 1 + 8.51iT - 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 + 15.3iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 13.4iT - 97T^{2} \) |
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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
−10.57350908731239145153630599142, −9.611884022627354372403543022072, −9.145565358490800521737378010825, −7.70529044987867028768695249597, −7.25549654138229578244611706389, −6.28899622935294998203927862791, −5.18807823905915805433305791122, −3.85735475014325986404458960568, −2.31734154628639968216076160824, −0.799521571847183911392092541634,
1.10027247022664126195243664359, 2.62346636959426063065352770316, 4.05528042263026462920474087232, 5.87123892464566269895121110986, 6.01357109097480799142440482788, 7.27685471614451363071515486156, 8.358744065000573032624858142218, 8.821539298158027074927332579530, 9.973471925001616319216447697484, 10.78417648825096905461549113789