L(s) = 1 | + (−0.835 − 0.549i)2-s + (−0.286 + 0.957i)3-s + (0.396 + 0.918i)4-s + (−0.686 − 0.727i)5-s + (0.766 − 0.642i)6-s + (−1.93 + 0.225i)7-s + (0.173 − 0.984i)8-s + (−0.835 − 0.549i)9-s + (0.173 + 0.984i)10-s + (−0.993 + 0.116i)12-s + (1.73 + 0.873i)14-s + (0.893 − 0.448i)15-s + (−0.686 + 0.727i)16-s + (0.396 + 0.918i)18-s + (0.396 − 0.918i)20-s + (0.337 − 1.91i)21-s + ⋯ |
L(s) = 1 | + (−0.835 − 0.549i)2-s + (−0.286 + 0.957i)3-s + (0.396 + 0.918i)4-s + (−0.686 − 0.727i)5-s + (0.766 − 0.642i)6-s + (−1.93 + 0.225i)7-s + (0.173 − 0.984i)8-s + (−0.835 − 0.549i)9-s + (0.173 + 0.984i)10-s + (−0.993 + 0.116i)12-s + (1.73 + 0.873i)14-s + (0.893 − 0.448i)15-s + (−0.686 + 0.727i)16-s + (0.396 + 0.918i)18-s + (0.396 − 0.918i)20-s + (0.337 − 1.91i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3998493356\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3998493356\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.835 + 0.549i)T \) |
| 3 | \( 1 + (0.286 - 0.957i)T \) |
| 5 | \( 1 + (0.686 + 0.727i)T \) |
good | 7 | \( 1 + (1.93 - 0.225i)T + (0.973 - 0.230i)T^{2} \) |
| 11 | \( 1 + (0.835 - 0.549i)T^{2} \) |
| 13 | \( 1 + (0.993 - 0.116i)T^{2} \) |
| 17 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 19 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-1.97 - 0.230i)T + (0.973 + 0.230i)T^{2} \) |
| 29 | \( 1 + (-1.06 + 0.536i)T + (0.597 - 0.802i)T^{2} \) |
| 31 | \( 1 + (0.286 + 0.957i)T^{2} \) |
| 37 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 41 | \( 1 + (-0.0971 + 0.0639i)T + (0.396 - 0.918i)T^{2} \) |
| 43 | \( 1 + (-0.337 - 0.0800i)T + (0.893 + 0.448i)T^{2} \) |
| 47 | \( 1 + (0.997 + 1.34i)T + (-0.286 + 0.957i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.835 + 0.549i)T^{2} \) |
| 61 | \( 1 + (0.227 - 0.526i)T + (-0.686 - 0.727i)T^{2} \) |
| 67 | \( 1 + (-1.59 - 0.802i)T + (0.597 + 0.802i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (-0.396 - 0.918i)T^{2} \) |
| 83 | \( 1 + (-0.0971 - 0.0639i)T + (0.396 + 0.918i)T^{2} \) |
| 89 | \( 1 + (0.290 - 1.64i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
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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
−9.628442015011081777831760843420, −8.919379173265703643167113021247, −8.440715821435520866056556041476, −7.15739156994984201286928022182, −6.49024940080154300178253160350, −5.37156569365663473799324078386, −4.26613840425811950377840899494, −3.42823409821920735365810329626, −2.82666686030713405784636584801, −0.65052215094857398945467890743,
0.806980295074838910009534645652, 2.62062768415505633904708366932, 3.29408575154426993968555910888, 4.94213306866003155188230756795, 6.18681253944679365603120403082, 6.59197927490147749358719635066, 7.10833337183415168877425639869, 7.81762119485361929366208780084, 8.778871710083851229681096340029, 9.490014225100312902134887407839