| L(s) = 1 | + (1.88 + 2.36i)2-s + (−5.83 + 2.81i)3-s + (−0.260 + 1.13i)4-s + (−12.6 + 6.09i)5-s + (−17.6 − 8.50i)6-s + (−3.94 + 18.0i)7-s + (18.6 − 8.97i)8-s + (9.31 − 11.6i)9-s + (−38.2 − 18.4i)10-s + (1.11 + 1.40i)11-s + (−1.68 − 7.38i)12-s + (19.6 + 24.6i)13-s + (−50.2 + 24.8i)14-s + (56.6 − 71.0i)15-s + (64.8 + 31.2i)16-s + (3.96 + 17.3i)17-s + ⋯ |
| L(s) = 1 | + (0.667 + 0.837i)2-s + (−1.12 + 0.540i)3-s + (−0.0325 + 0.142i)4-s + (−1.13 + 0.544i)5-s + (−1.20 − 0.578i)6-s + (−0.212 + 0.977i)7-s + (0.823 − 0.396i)8-s + (0.345 − 0.432i)9-s + (−1.21 − 0.583i)10-s + (0.0306 + 0.0384i)11-s + (−0.0405 − 0.177i)12-s + (0.419 + 0.525i)13-s + (−0.959 + 0.474i)14-s + (0.975 − 1.22i)15-s + (1.01 + 0.488i)16-s + (0.0565 + 0.247i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.145740 + 0.962196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.145740 + 0.962196i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (3.94 - 18.0i)T \) |
| good | 2 | \( 1 + (-1.88 - 2.36i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (5.83 - 2.81i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (12.6 - 6.09i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 1.40i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-19.6 - 24.6i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-3.96 - 17.3i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 - 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + (44.3 - 194. i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (61.3 + 268. i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-7.15 - 31.3i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (294. - 141. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-405. - 195. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (227. + 285. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (91.3 - 400. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (-753. - 362. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-1.60 - 7.03i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 - 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (159. - 699. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-341. + 428. i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 - 195.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-801. + 1.00e3i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (378. - 474. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 - 665.T + 9.12e5T^{2} \) |
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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
−15.64502110235931498852485810780, −14.98739385595416809656604687699, −13.53866199900298719503193924331, −11.85831188447278928628075499559, −11.31095234330484767796849777761, −9.865080963006929778449386677898, −7.83553167481656198112545061357, −6.39353243654275198420985999544, −5.42530592666347260142732592855, −3.99763202015455250463416825851,
0.72235578538241783759695500101, 3.66744826637582117436631560895, 4.98130365391546669877365438642, 6.90149556713670426348559776895, 8.163663568533916155584562340107, 10.56099652145871230109118189775, 11.34530476694917110731729148949, 12.34072261043912725198630574050, 12.84861920644446549030998567342, 14.20043683791754679431528162349
