L(s) = 1 | + (−1.36 + 2.37i)3-s + (−0.238 − 0.413i)5-s + (7.63 − 13.2i)7-s + (9.74 + 16.8i)9-s + (24.8 + 42.9i)11-s + (4.05 + 7.02i)13-s + 1.30·15-s + (−25.1 + 43.5i)17-s + (−81.6 + 141. i)19-s + (20.8 + 36.1i)21-s + 200.·23-s + (62.3 − 108. i)25-s − 127.·27-s + 90.6·29-s + (−4.76 + 172. i)31-s + ⋯ |
L(s) = 1 | + (−0.263 + 0.456i)3-s + (−0.0213 − 0.0369i)5-s + (0.412 − 0.713i)7-s + (0.361 + 0.625i)9-s + (0.680 + 1.17i)11-s + (0.0865 + 0.149i)13-s + 0.0225·15-s + (−0.358 + 0.621i)17-s + (−0.986 + 1.70i)19-s + (0.217 + 0.376i)21-s + 1.81·23-s + (0.499 − 0.864i)25-s − 0.907·27-s + 0.580·29-s + (−0.0275 + 0.999i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.28575 + 0.826016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28575 + 0.826016i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 31 | \( 1 + (4.76 - 172. i)T \) |
good | 3 | \( 1 + (1.36 - 2.37i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (0.238 + 0.413i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-7.63 + 13.2i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-24.8 - 42.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-4.05 - 7.02i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (25.1 - 43.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (81.6 - 141. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 - 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90.6T + 2.43e4T^{2} \) |
| 37 | \( 1 + (-105. + 183. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (131. + 228. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-42.2 + 73.2i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 19.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (229. + 396. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-34.1 + 59.2i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + 573.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-446. - 773. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (487. + 844. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (475. + 823. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-149. + 259. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-133. - 231. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 901.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 29.7T + 9.12e5T^{2} \) |
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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
−13.01056948701272088874061781338, −12.14170688163995281669782149413, −10.70742053636373136266037865940, −10.33567175814568510577943494628, −8.933067986189251472293408145011, −7.65600227994254575911285542721, −6.55234096605932899827467920393, −4.86264492806207511913739963500, −4.03173033948351419819345584826, −1.69396607445187412624788288372,
0.924593300475784502193222957446, 2.93658812481736647214070432391, 4.75736660872937620733123680479, 6.17660683049678681764777176319, 7.06206274965311867817405005406, 8.649648250631638189860235307417, 9.289945081842506073367355222439, 11.12442765276701762483695769692, 11.52482298479584481354953886558, 12.81531914850111810908125777723