| L(s) = 1 | + (−2.71 − 0.237i)2-s + (1.47 + 0.914i)3-s + (5.34 + 0.942i)4-s + (1.08 + 1.95i)5-s + (−3.77 − 2.83i)6-s + (−1.29 − 0.907i)7-s + (−9.01 − 2.41i)8-s + (1.32 + 2.69i)9-s + (−2.47 − 5.56i)10-s + (0.162 − 0.447i)11-s + (6.99 + 6.27i)12-s + (0.242 + 2.76i)13-s + (3.30 + 2.77i)14-s + (−0.197 + 3.86i)15-s + (13.7 + 4.99i)16-s + (3.35 − 0.898i)17-s + ⋯ |
| L(s) = 1 | + (−1.91 − 0.167i)2-s + (0.849 + 0.527i)3-s + (2.67 + 0.471i)4-s + (0.484 + 0.875i)5-s + (−1.54 − 1.15i)6-s + (−0.489 − 0.343i)7-s + (−3.18 − 0.854i)8-s + (0.442 + 0.896i)9-s + (−0.782 − 1.76i)10-s + (0.0491 − 0.134i)11-s + (2.02 + 1.81i)12-s + (0.0671 + 0.767i)13-s + (0.882 + 0.740i)14-s + (−0.0509 + 0.998i)15-s + (3.42 + 1.24i)16-s + (0.813 − 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.582300 + 0.302863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582300 + 0.302863i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.47 - 0.914i)T \) |
| 5 | \( 1 + (-1.08 - 1.95i)T \) |
| good | 2 | \( 1 + (2.71 + 0.237i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (1.29 + 0.907i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.162 + 0.447i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.242 - 2.76i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-3.35 + 0.898i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.97 - 1.13i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.230 + 0.329i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-4.29 + 3.60i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.908 + 5.15i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.387 + 1.44i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (4.99 - 5.95i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.501 + 1.07i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-6.94 + 9.92i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (0.274 - 0.274i)T - 53iT^{2} \) |
| 59 | \( 1 + (3.33 - 1.21i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.59 + 9.05i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-13.1 + 1.15i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (13.8 + 7.97i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.07 + 7.73i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.85 + 3.40i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.253 + 2.90i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-3.50 - 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.86 - 3.19i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.54109294410618502953801053045, −11.85718373546551113853079679776, −10.74288332942457461299951280945, −10.01477053438273072843193182065, −9.450066529785390516176468084546, −8.331572878618652202417074921372, −7.33037278729247826425883421481, −6.33399082687422236311804168868, −3.40578164546308793868752719381, −2.13714731460980926635017694988,
1.27021849479135292831868957628, 2.80343839783072273804916714559, 5.87160637496006207618675030158, 7.02030782387352808340214555430, 8.180132398018682756827686699386, 8.784259694757823409276993527055, 9.644546026790894378121725142730, 10.45165148847228228965312252381, 12.11329134678103177367113519833, 12.75219876828963128054981291770
