L(s) = 1 | + (2.69 + 4.67i)2-s + (−10.5 + 18.3i)4-s + (−7.78 − 13.4i)5-s − 71.0·8-s + (42.0 − 72.8i)10-s + (15.9 − 27.6i)11-s + 72.5·13-s + (−107. − 185. i)16-s + (14.5 − 25.1i)17-s + (−54.4 − 94.2i)19-s + 329.·20-s + 172.·22-s + (−27.6 − 47.8i)23-s + (−58.8 + 101. i)25-s + (195. + 339. i)26-s + ⋯ |
L(s) = 1 | + (0.954 + 1.65i)2-s + (−1.32 + 2.29i)4-s + (−0.696 − 1.20i)5-s − 3.14·8-s + (1.32 − 2.30i)10-s + (0.438 − 0.758i)11-s + 1.54·13-s + (−1.67 − 2.90i)16-s + (0.207 − 0.358i)17-s + (−0.657 − 1.13i)19-s + 3.68·20-s + 1.67·22-s + (−0.250 − 0.434i)23-s + (−0.470 + 0.815i)25-s + (1.47 + 2.56i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.163831326\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.163831326\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-2.69 - 4.67i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (7.78 + 13.4i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-15.9 + 27.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 72.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-14.5 + 25.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (54.4 + 94.2i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (27.6 + 47.8i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 17.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + (28.0 - 48.6i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-147. - 256. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 16.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + (255. + 443. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-132. + 229. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-127. + 220. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-36.4 - 63.0i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-253. + 438. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 827.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-186. + 322. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (514. + 890. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 453.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (166. + 287. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−11.18557296381070102236412216347, −9.223401637188842476342444848850, −8.480725946790722484997256314228, −8.152977304223016670075455099360, −6.84954009997462670140087369187, −6.07866583705544673467649711080, −5.06737303833745900940267545629, −4.25718487891637797769291346491, −3.40996788099023983824793251898, −0.57090732941961906472623080822,
1.33514901140711926221419209742, 2.55126807161892881641000170073, 3.84484811813453565805671993835, 3.97933253449981881522820849353, 5.69667892330616958219396561188, 6.51220270944214406371047405588, 7.953707170067973160816913144875, 9.273099542039356658648320202445, 10.18772138942700314446819881353, 10.97901608677837350383219738361