| L(s) = 1 | + (−0.376 + 1.36i)2-s + (−1.71 − 1.02i)4-s − 1.04·7-s + (2.04 − 1.95i)8-s + 2.67i·11-s + 0.761·13-s + (0.392 − 1.41i)14-s + (1.89 + 3.52i)16-s + 2.17·17-s − 3.36·19-s + (−3.64 − 1.00i)22-s − 4.98i·23-s + (−0.286 + 1.03i)26-s + (1.78 + 1.06i)28-s + 7.88·29-s + ⋯ |
| L(s) = 1 | + (−0.266 + 0.963i)2-s + (−0.858 − 0.513i)4-s − 0.393·7-s + (0.723 − 0.690i)8-s + 0.807i·11-s + 0.211·13-s + (0.104 − 0.379i)14-s + (0.473 + 0.880i)16-s + 0.528·17-s − 0.772·19-s + (−0.777 − 0.214i)22-s − 1.03i·23-s + (−0.0562 + 0.203i)26-s + (0.337 + 0.202i)28-s + 1.46·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9868288593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9868288593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.376 - 1.36i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 - 0.761T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 3.36T + 19T^{2} \) |
| 23 | \( 1 + 4.98iT - 23T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 31 | \( 1 - 4.48iT - 31T^{2} \) |
| 37 | \( 1 - 0.663T + 37T^{2} \) |
| 41 | \( 1 - 8.94iT - 41T^{2} \) |
| 43 | \( 1 - 2.41iT - 43T^{2} \) |
| 47 | \( 1 + 7.27iT - 47T^{2} \) |
| 53 | \( 1 + 1.00iT - 53T^{2} \) |
| 59 | \( 1 - 4.94iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 14.3iT - 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 - 3.65iT - 73T^{2} \) |
| 79 | \( 1 - 13.3iT - 79T^{2} \) |
| 83 | \( 1 + 7.77T + 83T^{2} \) |
| 89 | \( 1 - 6.69iT - 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−9.564536252761136532608521840448, −8.518054582915755928539675438384, −8.180021680977741814047433805226, −6.97780677259355207881864775012, −6.64412307556901356466764726699, −5.70368423827857135862929355872, −4.75662291375090737396757402641, −4.07159585996487186737495306825, −2.72437631268768802542411965827, −1.18360150220920601923174448850,
0.46262109207472617510754591347, 1.78424522857323489150766553998, 2.98727701576838852864921653703, 3.64902441066322028654136403801, 4.64790250696797533912188755932, 5.65291837743178043976563877480, 6.52202904916145341646572698897, 7.71038332082028560145019797823, 8.303696601394391429712639614345, 9.139315965281729976566133781982
