L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (8.00 − 13.8i)5-s + (1.70 − 18.4i)7-s − 7.99·8-s + (−16.0 − 27.7i)10-s + (−7.89 − 13.6i)11-s − 50.4·13-s + (−30.2 − 21.4i)14-s + (−8 + 13.8i)16-s + (18.4 + 32.0i)17-s + (44.6 − 77.4i)19-s − 64.0·20-s − 31.5·22-s + (−30.6 + 53.1i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.716 − 1.24i)5-s + (0.0922 − 0.995i)7-s − 0.353·8-s + (−0.506 − 0.877i)10-s + (−0.216 − 0.374i)11-s − 1.07·13-s + (−0.577 − 0.408i)14-s + (−0.125 + 0.216i)16-s + (0.263 + 0.456i)17-s + (0.539 − 0.934i)19-s − 0.716·20-s − 0.306·22-s + (−0.278 + 0.481i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.948996648\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.948996648\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.70 + 18.4i)T \) |
good | 5 | \( 1 + (-8.00 + 13.8i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (7.89 + 13.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 50.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-18.4 - 32.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-44.6 + 77.4i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (30.6 - 53.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 126.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-129. - 225. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-123. + 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 516.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + (182. - 316. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-23.5 - 40.8i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (358. + 620. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-348. + 603. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (55.3 + 95.8i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 607.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-140. - 242. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-441. + 764. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 63.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-337. + 585. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 976.T + 9.12e5T^{2} \) |
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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
−10.36248731848865956667640762284, −9.757421151622895103417726986117, −8.837961907681552891569481987701, −7.77036009016216060621119123716, −6.48443922562348461158271404526, −5.13891382667963143889537145158, −4.69460247562870259117520781086, −3.23407876064048439419595253761, −1.68911399444701172437930817569, −0.57686313330881880613319369787,
2.24534000652016275993402283988, 3.08040547777373522846601859957, 4.77087438290865956155777571270, 5.75225967812683539657255920725, 6.55231421351646382584671014642, 7.46980426356711314634414865434, 8.460837590551585558255544457213, 9.838013815623014901374937924035, 10.11867589486905334608677881882, 11.68677128289464211881384807649