| L(s) = 1 | + (−4.26 + 2.46i)2-s + (−1.44 − 4.99i)3-s + (8.14 − 14.1i)4-s − 9.24·5-s + (18.4 + 17.7i)6-s + (17.8 + 4.75i)7-s + 40.8i·8-s + (−22.8 + 14.3i)9-s + (39.4 − 22.7i)10-s + 15.2i·11-s + (−82.1 − 20.3i)12-s + (−61.0 + 35.2i)13-s + (−88.1 + 23.7i)14-s + (13.3 + 46.1i)15-s + (−35.5 − 61.5i)16-s + (17.8 + 30.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.50 + 0.871i)2-s + (−0.277 − 0.960i)3-s + (1.01 − 1.76i)4-s − 0.827·5-s + (1.25 + 1.20i)6-s + (0.966 + 0.256i)7-s + 1.80i·8-s + (−0.845 + 0.533i)9-s + (1.24 − 0.720i)10-s + 0.418i·11-s + (−1.97 − 0.488i)12-s + (−1.30 + 0.752i)13-s + (−1.68 + 0.454i)14-s + (0.229 + 0.794i)15-s + (−0.555 − 0.961i)16-s + (0.255 + 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.170623 + 0.295999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170623 + 0.295999i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.44 + 4.99i)T \) |
| 7 | \( 1 + (-17.8 - 4.75i)T \) |
| good | 2 | \( 1 + (4.26 - 2.46i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 9.24T + 125T^{2} \) |
| 11 | \( 1 - 15.2iT - 1.33e3T^{2} \) |
| 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 | \( 1 - 92.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-222. - 128. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (123. + 71.3i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 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 | \( 1 + (235. + 408. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (247. - 142. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-85.0 + 147. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-222. + 128. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (501. - 868. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 445. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (81.3 - 46.9i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (231. + 401. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 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
−15.04248897529261248643533353469, −14.17684113474359484469747832716, −12.15334412835020432112935731201, −11.47031818541984406154788589257, −9.991144048407612348530736975720, −8.538171788941609258249813877699, −7.67265944773517196471869684539, −6.99923422984235898452104129329, −5.31993248589260663745898265188, −1.60119749586496727353484805997,
0.40604007670389094664067130990, 3.03782020462618467689624349687, 4.84712600057070141349064609192, 7.51607105680032147495069220021, 8.411953492600408832091238970799, 9.660588913698121647610104486075, 10.58210838058522750677524818153, 11.50294665408444529983568368562, 12.11722899173328059202892809959, 14.35516552500054165871322492562
