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
\(L(2)\) |
\(\approx\) |
\(1.011023380\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011023380\) |
\(L(\frac{5}{2})\) |
|
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
−11.23605477161504199018082742376, −9.761416498923706114011744213613, −9.203968329697177204014082720413, −8.251011846351927019330803111096, −7.66718642315709133639249986275, −5.92004538944911059551076229761, −4.89035770542477010951312591479, −3.86999114301430309552949000768, −2.40246472371457043392049433565, −0.957360864860469165726369789507,
0.51422186877649847411972184561, 2.61968211464236578098293686241, 3.97335346672927476424184483264, 5.02436463979391563711410775962, 6.64500337473951383737464926251, 7.21170108384357576032515045761, 7.70823782598669347944607739786, 9.117008112509984092065336005115, 10.05659719168898427638075979609, 10.84963297767316778104969555654