L(s) = 1 | + (−1.30 + 0.951i)5-s + (0.0729 + 0.224i)7-s + (0.809 + 3.21i)11-s + (3.42 + 2.48i)13-s + (−0.118 + 0.0857i)17-s + (−2.11 + 6.51i)19-s + 5·23-s + (−0.736 + 2.26i)25-s + (−0.618 − 1.90i)29-s + (−2.73 − 1.98i)31-s + (−0.309 − 0.224i)35-s + (0.690 + 2.12i)37-s + (2.69 − 8.28i)41-s − 2.52·43-s + (−2.95 + 9.09i)47-s + ⋯ |
L(s) = 1 | + (−0.585 + 0.425i)5-s + (0.0275 + 0.0848i)7-s + (0.243 + 0.969i)11-s + (0.950 + 0.690i)13-s + (−0.0286 + 0.0207i)17-s + (−0.485 + 1.49i)19-s + 1.04·23-s + (−0.147 + 0.453i)25-s + (−0.114 − 0.353i)29-s + (−0.491 − 0.357i)31-s + (−0.0522 − 0.0379i)35-s + (0.113 + 0.349i)37-s + (0.420 − 1.29i)41-s − 0.385·43-s + (−0.431 + 1.32i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.974536 + 0.675816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.974536 + 0.675816i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-0.809 - 3.21i)T \) |
good | 5 | \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.0729 - 0.224i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.42 - 2.48i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.118 - 0.0857i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.11 - 6.51i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 + (0.618 + 1.90i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.73 + 1.98i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.690 - 2.12i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.69 + 8.28i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 + (2.95 - 9.09i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.35 - 3.88i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (3.64 + 11.2i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-8.78 + 6.37i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 + (11.7 - 8.55i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.85 + 14.9i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.04 - 5.11i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.42 - 1.03i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 5.18T + 89T^{2} \) |
| 97 | \( 1 + (12.1 + 8.83i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.42779751903833990524404918715, −10.68124137215923633698081461041, −9.644119042258777059322869283600, −8.715199407439125973517940901249, −7.67347735837214704393782417591, −6.84844712028637821353997263443, −5.80361438710434876042063661569, −4.37647637538400691206784596825, −3.50423525813854271388238601978, −1.80853942679714421666427884745,
0.842577288317813414605117291850, 2.95794669535776452074243716417, 4.08871117031297673956140307819, 5.24666660033984875796236567599, 6.35431753915924871455654508631, 7.42930384808506014445648346910, 8.600570512594095172804076570627, 8.907206882097210255747216555563, 10.41798327317094666871518986704, 11.15384611041681204722246047447