| L(s) = 1 | + (2.52 + 1.45i)2-s + (−4.29 − 2.92i)3-s + (0.247 + 0.428i)4-s + 17.2·5-s + (−6.57 − 13.6i)6-s + (17.7 + 5.37i)7-s − 21.8i·8-s + (9.88 + 25.1i)9-s + (43.4 + 25.0i)10-s − 35.6i·11-s + (0.191 − 2.56i)12-s + (−8.99 − 5.19i)13-s + (36.9 + 39.3i)14-s + (−73.9 − 50.3i)15-s + (33.8 − 58.6i)16-s + (−64.1 + 111. i)17-s + ⋯ |
| L(s) = 1 | + (0.892 + 0.515i)2-s + (−0.826 − 0.562i)3-s + (0.0309 + 0.0536i)4-s + 1.53·5-s + (−0.447 − 0.928i)6-s + (0.956 + 0.290i)7-s − 0.966i·8-s + (0.366 + 0.930i)9-s + (1.37 + 0.793i)10-s − 0.978i·11-s + (0.00459 − 0.0617i)12-s + (−0.191 − 0.110i)13-s + (0.704 + 0.752i)14-s + (−1.27 − 0.866i)15-s + (0.529 − 0.916i)16-s + (−0.915 + 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.11224 - 0.150069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11224 - 0.150069i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.29 + 2.92i)T \) |
| 7 | \( 1 + (-17.7 - 5.37i)T \) |
| good | 2 | \( 1 + (-2.52 - 1.45i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 11 | \( 1 + 35.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (8.99 + 5.19i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (64.1 - 111. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (84.7 - 48.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 - 25.5iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (14.1 - 8.16i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-160. + 92.7i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (0.462 + 0.801i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (231. - 401. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (43.2 + 74.9i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (143. - 248. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (334. + 193. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (45.2 + 78.3i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-120. - 69.7i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (151. + 262. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 351. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-761. - 439. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-7.12 + 12.3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (125. + 217. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (505. + 875. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.12e3 + 649. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.23016282030649187018786086568, −13.38737039065595649738318125788, −12.68580124081708655883080504186, −11.11483651871246029730614329453, −10.07985619343851748877394618165, −8.336841638635205369824372363093, −6.34191869366165377203902671587, −5.92209682757307712397136272913, −4.72938966564114353997904539634, −1.70346353631031498184091056040,
2.19806493526229870555852481794, 4.57131567163599267777925355793, 5.17592435468955689049514771962, 6.76541446251182790347577315342, 8.993404242651912150555266615208, 10.21626846961188667497925949698, 11.21820126347599560383483237475, 12.25920829385873828616832505395, 13.39679060852958267010587219410, 14.20663658328767459487758286825
