| L(s) = 1 | + (6.12 − 5.14i)2-s + (26.6 + 9.69i)3-s + (11.1 − 63.0i)4-s + (−62.1 − 352. i)5-s + (213. − 77.5i)6-s + (−141. + 245. i)7-s + (−256. − 443. i)8-s + (−1.05e3 − 889. i)9-s + (−2.19e3 − 1.83e3i)10-s + (−926. − 1.60e3i)11-s + (907. − 1.57e3i)12-s + (8.08e3 − 2.94e3i)13-s + (393. + 2.23e3i)14-s + (1.76e3 − 9.98e3i)15-s + (−3.84e3 − 1.40e3i)16-s + (3.61e3 − 3.03e3i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.569 + 0.207i)3-s + (0.0868 − 0.492i)4-s + (−0.222 − 1.26i)5-s + (0.402 − 0.146i)6-s + (−0.156 + 0.270i)7-s + (−0.176 − 0.306i)8-s + (−0.484 − 0.406i)9-s + (−0.693 − 0.581i)10-s + (−0.209 − 0.363i)11-s + (0.151 − 0.262i)12-s + (1.02 − 0.371i)13-s + (0.0383 + 0.217i)14-s + (0.134 − 0.764i)15-s + (−0.234 − 0.0855i)16-s + (0.178 − 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.24767 - 2.01326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24767 - 2.01326i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-6.12 + 5.14i)T \) |
| 19 | \( 1 + (2.43e4 + 1.73e4i)T \) |
| good | 3 | \( 1 + (-26.6 - 9.69i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (62.1 + 352. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (141. - 245. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (926. + 1.60e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-8.08e3 + 2.94e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-3.61e3 + 3.03e3i)T + (7.12e7 - 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-3.90e3 + 2.21e4i)T + (-3.19e9 - 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.87e5 - 1.56e5i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (1.75e4 - 3.03e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-4.78e5 - 1.74e5i)T + (1.49e11 + 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-3.02e4 - 1.71e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-3.53e5 - 2.96e5i)T + (8.79e10 + 4.98e11i)T^{2} \) |
| 53 | \( 1 + (1.08e5 - 6.14e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-7.37e5 + 6.18e5i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-5.29e4 + 3.00e5i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-1.31e5 - 1.10e5i)T + (1.05e12 + 5.96e12i)T^{2} \) |
| 71 | \( 1 + (9.38e5 + 5.31e6i)T + (-8.54e12 + 3.11e12i)T^{2} \) |
| 73 | \( 1 + (4.02e6 + 1.46e6i)T + (8.46e12 + 7.10e12i)T^{2} \) |
| 79 | \( 1 + (1.63e6 + 5.96e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (3.90e6 - 6.76e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.12e7 + 4.10e6i)T + (3.38e13 - 2.84e13i)T^{2} \) |
| 97 | \( 1 + (1.52e6 - 1.27e6i)T + (1.40e13 - 7.95e13i)T^{2} \) |
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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
−14.25307211532894835670733915777, −13.09364495878563784584543899118, −12.22893587570848733811073053752, −10.85262999141603486769728083753, −9.141366114278274544745018234894, −8.402135899905637507632786049137, −6.00598459348989537036128459343, −4.50560250382341823658164556658, −2.98342678045736836047341470177, −0.845659541946209000447883729021,
2.51225126101711480433374153046, 3.89084279812613567863444591076, 6.06757374801093617806237210942, 7.29718271895207796345468726234, 8.451958232572321156084751156585, 10.39319651650716725204794449673, 11.55106868667800007034166537638, 13.19248144716289264903097411022, 14.12446701560477472029082154922, 14.89659622880981507767166824407
