L(s) = 1 | + (−1 − 1.41i)3-s + (0.5 + 0.866i)5-s + (1.72 − 2.98i)7-s + (−1.00 + 2.82i)9-s + (0.724 − 1.25i)11-s + (−2.94 − 5.10i)13-s + (0.724 − 1.57i)15-s + 4.89·17-s − 4·19-s + (−5.94 + 0.548i)21-s + (−2.72 − 4.71i)23-s + (2 − 3.46i)25-s + (5.00 − 1.41i)27-s + (−0.0505 + 0.0874i)29-s + (1.27 + 2.20i)31-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.816i)3-s + (0.223 + 0.387i)5-s + (0.651 − 1.12i)7-s + (−0.333 + 0.942i)9-s + (0.218 − 0.378i)11-s + (−0.818 − 1.41i)13-s + (0.187 − 0.406i)15-s + 1.18·17-s − 0.917·19-s + (−1.29 + 0.119i)21-s + (−0.568 − 0.984i)23-s + (0.400 − 0.692i)25-s + (0.962 − 0.272i)27-s + (−0.00937 + 0.0162i)29-s + (0.229 + 0.396i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00922 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.770666 - 0.763586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.770666 - 0.763586i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.72 + 2.98i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.724 + 1.25i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.94 + 5.10i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (2.72 + 4.71i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0505 - 0.0874i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.27 - 2.20i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.898T + 37T^{2} \) |
| 41 | \( 1 + (-5.94 - 10.3i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.17 - 2.03i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.17 - 5.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.89T + 53T^{2} \) |
| 59 | \( 1 + (-7.17 - 12.4i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.94 + 6.84i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.17 - 10.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.79T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.275 - 0.476i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + (-1.94 + 3.37i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.59215700464073073998187706270, −10.48728265080686680979649596676, −10.25454398613485757442174198957, −8.277808320956307344423857159326, −7.69232019076556994287757704351, −6.68910171845857904948459346737, −5.67762622913793743669812234480, −4.48761440104941814252176211708, −2.72527788643027124441115982017, −0.929514809948609905241792156450,
2.02961774183538419465065363840, 3.94352999670295146950212585372, 5.06435268804208249331100137762, 5.73387370260748724227880405511, 7.05741419137778495630014040907, 8.528494844718235248589158491172, 9.321572151353863595765705943370, 10.00769454390008125829801915692, 11.30434519806985104988906977288, 11.93998518842179630902278668328