| L(s) = 1 | + 2·2-s + (−5.16 + 0.531i)3-s + 4·4-s + (1.73 − 2.99i)5-s + (−10.3 + 1.06i)6-s + (−7.22 + 17.0i)7-s + 8·8-s + (26.4 − 5.49i)9-s + (3.46 − 5.99i)10-s + (34.1 + 59.2i)11-s + (−20.6 + 2.12i)12-s + (11.0 + 19.1i)13-s + (−14.4 + 34.1i)14-s + (−7.35 + 16.4i)15-s + 16·16-s + (−8.70 + 15.0i)17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (−0.994 + 0.102i)3-s + 0.5·4-s + (0.154 − 0.268i)5-s + (−0.703 + 0.0723i)6-s + (−0.390 + 0.920i)7-s + 0.353·8-s + (0.979 − 0.203i)9-s + (0.109 − 0.189i)10-s + (0.937 + 1.62i)11-s + (−0.497 + 0.0511i)12-s + (0.235 + 0.408i)13-s + (−0.275 + 0.651i)14-s + (−0.126 + 0.282i)15-s + 0.250·16-s + (−0.124 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.420 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.48812 + 0.950849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48812 + 0.950849i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (5.16 - 0.531i)T \) |
| 7 | \( 1 + (7.22 - 17.0i)T \) |
| good | 5 | \( 1 + (-1.73 + 2.99i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-34.1 - 59.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-11.0 - 19.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (8.70 - 15.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (12.4 + 21.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (56.2 - 97.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-50.9 + 88.3i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 174.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (103. + 178. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (182. + 316. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-131. + 227. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 4.38T + 1.03e5T^{2} \) |
| 53 | \( 1 + (254. - 440. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + 541.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 282.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 842.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 249.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (109. - 189. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 335.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (514. - 890. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (386. + 669. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-900. + 1.55e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.70518449383775061516113550560, −12.21199111557669161299992581386, −11.40271457649331062657326704572, −10.03332764323242017117804320967, −9.122483742491452121085258111018, −7.16975930800492235902580864351, −6.25485508839897890208427553130, −5.16097133901812222886133253968, −4.04303171844725549161782642421, −1.83923067264234382536985926409,
0.887706594152582383850012037299, 3.36779271034175581133032193145, 4.66452222018142857509941735266, 6.23758248413234040120352788876, 6.58526137558439061957401752364, 8.212774163551015911330857065630, 10.03708867866073417936843150380, 10.85698489678837444718658624622, 11.66510864592301128154018980898, 12.74477047947880346550252875475
