| L(s) = 1 | + (−7.51 − 2.73i)2-s + (11.5 + 65.2i)3-s + (49.0 + 41.1i)4-s + (158. − 132. i)5-s + (92.0 − 521. i)6-s + (668. − 1.15e3i)7-s + (−256. − 443. i)8-s + (−2.06e3 + 753. i)9-s + (−1.55e3 + 565. i)10-s + (3.79e3 + 6.57e3i)11-s + (−2.12e3 + 3.67e3i)12-s + (1.97e3 − 1.11e4i)13-s + (−8.19e3 + 6.87e3i)14-s + (1.04e4 + 8.79e3i)15-s + (711. + 4.03e3i)16-s + (1.87e3 + 682. i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.246 + 1.39i)3-s + (0.383 + 0.321i)4-s + (0.566 − 0.475i)5-s + (0.173 − 0.986i)6-s + (0.737 − 1.27i)7-s + (−0.176 − 0.306i)8-s + (−0.946 + 0.344i)9-s + (−0.491 + 0.178i)10-s + (0.860 + 1.48i)11-s + (−0.354 + 0.613i)12-s + (0.249 − 1.41i)13-s + (−0.798 + 0.670i)14-s + (0.802 + 0.673i)15-s + (0.0434 + 0.246i)16-s + (0.0925 + 0.0336i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.70328 + 0.462138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70328 + 0.462138i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.51 + 2.73i)T \) |
| 19 | \( 1 + (-2.37e4 - 1.81e4i)T \) |
| good | 3 | \( 1 + (-11.5 - 65.2i)T + (-2.05e3 + 747. i)T^{2} \) |
| 5 | \( 1 + (-158. + 132. i)T + (1.35e4 - 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-668. + 1.15e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.79e3 - 6.57e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.97e3 + 1.11e4i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (-1.87e3 - 682. i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 23 | \( 1 + (5.80e3 + 4.86e3i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-1.85e5 + 6.74e4i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (1.22e5 - 2.12e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.41e4 + 1.93e5i)T + (-1.83e11 + 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-2.65e5 + 2.22e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (-7.83e5 + 2.84e5i)T + (3.88e11 - 3.25e11i)T^{2} \) |
| 53 | \( 1 + (3.48e4 + 2.92e4i)T + (2.03e11 + 1.15e12i)T^{2} \) |
| 59 | \( 1 + (2.41e5 + 8.78e4i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-8.83e5 - 7.41e5i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (1.27e6 - 4.63e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-3.93e5 + 3.29e5i)T + (1.57e12 - 8.95e12i)T^{2} \) |
| 73 | \( 1 + (5.59e5 + 3.17e6i)T + (-1.03e13 + 3.77e12i)T^{2} \) |
| 79 | \( 1 + (1.14e6 + 6.51e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-3.35e6 + 5.81e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (2.20e6 - 1.24e7i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (-1.64e6 - 5.98e5i)T + (6.18e13 + 5.19e13i)T^{2} \) |
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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
−15.04782947262501770343310302557, −13.93167856655614261177019027678, −12.25729198117483788307988950784, −10.51765736411144235684595615358, −10.10442072280924759047430481641, −8.909515405974345883237579612296, −7.41369383543736389127476754401, −5.02551104286068636228486338412, −3.72399708611865692009574537566, −1.31426510372647658925062814285,
1.26414178578457479263433809947, 2.45880905390146318266419121318, 5.89393341043515962745362915583, 6.81657947299470768380858139319, 8.342977843754956494733715403725, 9.164879975882293980993558908852, 11.28205297652075020431730838560, 12.03686014556037831658015904871, 13.84894575162166574778783333091, 14.30810140617212637876669669623
