L(s) = 1 | + (0.559 + 1.29i)2-s + (0.224 − 0.839i)3-s + (−1.37 + 1.45i)4-s + (0.847 + 3.16i)5-s + (1.21 − 0.177i)6-s + (−0.654 − 2.56i)7-s + (−2.65 − 0.968i)8-s + (1.94 + 1.12i)9-s + (−3.63 + 2.87i)10-s + (2.87 + 0.769i)11-s + (0.911 + 1.47i)12-s + (−3.63 − 3.63i)13-s + (2.96 − 2.28i)14-s + 2.84·15-s + (−0.229 − 3.99i)16-s + (−1.81 − 3.14i)17-s + ⋯ |
L(s) = 1 | + (0.395 + 0.918i)2-s + (0.129 − 0.484i)3-s + (−0.686 + 0.727i)4-s + (0.379 + 1.41i)5-s + (0.496 − 0.0726i)6-s + (−0.247 − 0.968i)7-s + (−0.939 − 0.342i)8-s + (0.648 + 0.374i)9-s + (−1.14 + 0.908i)10-s + (0.865 + 0.231i)11-s + (0.263 + 0.426i)12-s + (−1.00 − 1.00i)13-s + (0.791 − 0.610i)14-s + 0.734·15-s + (−0.0574 − 0.998i)16-s + (−0.441 − 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00741 + 0.755831i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00741 + 0.755831i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.559 - 1.29i)T \) |
| 7 | \( 1 + (0.654 + 2.56i)T \) |
good | 3 | \( 1 + (-0.224 + 0.839i)T + (-2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.847 - 3.16i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.87 - 0.769i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (3.63 + 3.63i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.81 + 3.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.60 + 0.429i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.33 + 3.08i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.10 - 5.10i)T + 29iT^{2} \) |
| 31 | \( 1 + (1.00 + 1.74i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.49 - 5.57i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 3.71iT - 41T^{2} \) |
| 43 | \( 1 + (2.91 - 2.91i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.06 - 8.77i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.68 - 0.986i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.64 + 0.977i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (6.54 - 1.75i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (1.57 - 5.88i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 9.55iT - 71T^{2} \) |
| 73 | \( 1 + (0.989 - 0.571i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.120 + 0.209i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.459 + 0.459i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.76 - 2.17i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−14.04880755635619671654366206837, −13.17523142915879432602334389652, −12.07582909019758829353244557632, −10.49470258886773283927995378247, −9.682053968887918908792258404911, −7.81497808526472623520597566645, −7.06842215851006500728995862973, −6.40988201763444123296502387692, −4.58005225605456781388041011019, −2.97254012049944524773591715810,
1.82674729320211907682559746413, 3.96004501770319414576577037629, 4.94717277177938885117712792300, 6.21530042535009233145114151201, 8.647625804148090726834729972565, 9.364310685597800733808352106879, 9.956318934586339450207488502347, 11.79395076454031676219094490308, 12.22350306214123826279588869459, 13.17411375399885183175577984355