L(s) = 1 | + 4.92i·2-s + (3.60 + 3.74i)3-s − 16.2·4-s + (−4.62 − 8.00i)5-s + (−18.4 + 17.7i)6-s + (−13.0 + 13.1i)7-s − 40.8i·8-s + (−1.04 + 26.9i)9-s + (39.4 − 22.7i)10-s + (13.2 + 7.62i)11-s + (−58.6 − 61.0i)12-s + (61.0 + 35.2i)13-s + (−64.6 − 64.4i)14-s + (13.3 − 46.1i)15-s + 71.0·16-s + (−17.8 − 30.9i)17-s + ⋯ |
L(s) = 1 | + 1.74i·2-s + (0.693 + 0.720i)3-s − 2.03·4-s + (−0.413 − 0.716i)5-s + (−1.25 + 1.20i)6-s + (−0.705 + 0.708i)7-s − 1.80i·8-s + (−0.0388 + 0.999i)9-s + (1.24 − 0.720i)10-s + (0.362 + 0.209i)11-s + (−1.41 − 1.46i)12-s + (1.30 + 0.752i)13-s + (−1.23 − 1.22i)14-s + (0.229 − 0.794i)15-s + 1.11·16-s + (−0.255 − 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.963 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.183317 - 1.33625i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.183317 - 1.33625i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.60 - 3.74i)T \) |
| 7 | \( 1 + (13.0 - 13.1i)T \) |
good | 2 | \( 1 - 4.92iT - 8T^{2} \) |
| 5 | \( 1 + (4.62 + 8.00i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-13.2 - 7.62i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-61.0 - 35.2i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (17.8 + 30.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-91.6 - 52.9i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (80.0 - 46.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-222. + 128. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 142. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (100. - 173. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-162. + 280. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (92.5 + 160. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 471.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-247. + 142. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 170.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 257. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 445. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (81.3 - 46.9i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 463.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-348. - 603. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-338. + 586. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (402. - 232. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.48096783642901426272708258050, −14.17494981569733620772415237621, −13.51541743889093187074878193213, −11.95158457174129282827444098282, −9.770215273832517454038322349583, −8.837599399033942712169725985745, −8.174327385768024242435480162931, −6.58188694613211172157941476548, −5.20534466885961021332858297450, −3.87889297453154010059920558523,
0.961531433778671484488781829017, 2.98564752038031549584017405310, 3.76620412730336584937729743124, 6.62250865333260432977702555837, 8.200482442004087583407371018944, 9.484933696180464461111915275620, 10.63707719205482679509464709540, 11.54385147940049778209312620479, 12.78356405990745669232894838420, 13.46604917321703539458675201963