L(s) = 1 | + (3.47 − 1.13i)2-s + (1.22 + 1.68i)3-s + (4.35 − 3.16i)4-s + (−11.1 + 0.0808i)5-s + (6.17 + 4.48i)6-s − 8.69i·7-s + (−5.63 + 7.75i)8-s + (6.99 − 21.5i)9-s + (−38.8 + 12.9i)10-s + (11.2 + 34.5i)11-s + (10.6 + 3.46i)12-s + (9.00 + 2.92i)13-s + (−9.82 − 30.2i)14-s + (−13.8 − 18.7i)15-s + (−24.1 + 74.2i)16-s + (14.7 − 20.3i)17-s + ⋯ |
L(s) = 1 | + (1.22 − 0.399i)2-s + (0.235 + 0.324i)3-s + (0.543 − 0.395i)4-s + (−0.999 + 0.00722i)5-s + (0.419 + 0.305i)6-s − 0.469i·7-s + (−0.249 + 0.342i)8-s + (0.259 − 0.797i)9-s + (−1.22 + 0.408i)10-s + (0.307 + 0.946i)11-s + (0.256 + 0.0834i)12-s + (0.192 + 0.0623i)13-s + (−0.187 − 0.577i)14-s + (−0.238 − 0.323i)15-s + (−0.377 + 1.16i)16-s + (0.210 − 0.290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.81064 - 0.215450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81064 - 0.215450i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (11.1 - 0.0808i)T \) |
good | 2 | \( 1 + (-3.47 + 1.13i)T + (6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (-1.22 - 1.68i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 8.69iT - 343T^{2} \) |
| 11 | \( 1 + (-11.2 - 34.5i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-9.00 - 2.92i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-14.7 + 20.3i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (43.5 + 31.6i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-186. + 60.5i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (153. - 111. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (201. + 146. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (192. + 62.5i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (148. - 458. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 13.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (223. + 307. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-354. - 487. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (121. - 373. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (127. + 391. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-244. + 336. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-259. + 188. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (141. - 45.9i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (29.9 - 21.7i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-629. + 866. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (4.60 + 14.1i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-709. - 976. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−16.82648284984280483612044557813, −15.13707812067254718215885325462, −14.78621035471824819764061521051, −13.10103321039883701154678920905, −12.16116039459300329338658716570, −10.94765082805903925733360107262, −9.016196566559580540371057046983, −7.01417290457198233509527103627, −4.65213313673574854748328179818, −3.51380931075340905272132934446,
3.57795676572230121182247705709, 5.34533078024026741064592251903, 7.11673014095781242999765141266, 8.667051879187692968384423580245, 11.06913601011922402944653362802, 12.42975007630864453318422208126, 13.41525632879261459690936080420, 14.62586111254217239113747734413, 15.59784492928327163852028331622, 16.63913351806243359307614787823