L(s) = 1 | + (−1.5 − 2.59i)3-s + (−9.47 + 16.4i)5-s + (12.8 − 13.3i)7-s + (−4.5 + 7.79i)9-s + (−27.3 − 47.3i)11-s + 62.0·13-s + 56.8·15-s + (−61.2 − 106. i)17-s + (6.25 − 10.8i)19-s + (−53.9 − 13.1i)21-s + (37.2 − 64.4i)23-s + (−117. − 203. i)25-s + 27·27-s − 232.·29-s + (−5.18 − 8.97i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.847 + 1.46i)5-s + (0.691 − 0.722i)7-s + (−0.166 + 0.288i)9-s + (−0.750 − 1.29i)11-s + 1.32·13-s + 0.978·15-s + (−0.873 − 1.51i)17-s + (0.0755 − 0.130i)19-s + (−0.560 − 0.137i)21-s + (0.337 − 0.584i)23-s + (−0.937 − 1.62i)25-s + 0.192·27-s − 1.48·29-s + (−0.0300 − 0.0520i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.620934 - 0.741352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.620934 - 0.741352i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 + (-12.8 + 13.3i)T \) |
good | 5 | \( 1 + (9.47 - 16.4i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (27.3 + 47.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 62.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + (61.2 + 106. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-6.25 + 10.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-37.2 + 64.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (5.18 + 8.97i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-122. + 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-242. + 420. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-189. - 327. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-91.3 - 158. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (198. - 343. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-130. - 226. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 874.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (76.2 + 131. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (286. - 496. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 317.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-47.5 + 82.2i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−11.59247149858468889252833483970, −11.06865184649860388951238526040, −10.66880889553210943785173141696, −8.718502763199744147126787275771, −7.62185629625713816015530341358, −6.98424429722705676126952196737, −5.74333340403877349469782524656, −4.00455925916610848761935904876, −2.74298192852873173424704292698, −0.48148941959570695218262568299,
1.60195402923005614836986799945, 3.99919149971048796991793690535, 4.82713014809901132334423929791, 5.86122206868806473572176012404, 7.75763791523220469491928184500, 8.548069614648120679031211198382, 9.362366581762346604689184795692, 10.82662729537695756453911667810, 11.60413890179999848695336035557, 12.64026365794903157314141766087