| L(s) = 1 | + (3.15 + 2.83i)3-s + (4.11 − 7.13i)5-s + (1.01 − 9.65i)7-s + (0.940 + 8.94i)9-s + (−0.316 − 0.711i)11-s + (−5.04 + 23.7i)13-s + (33.2 − 10.8i)15-s + (−10.1 + 22.7i)17-s + (−20.9 + 4.45i)19-s + (30.5 − 27.5i)21-s + (−1.53 − 2.11i)23-s + (−21.4 − 37.1i)25-s + (0.00995 − 0.0137i)27-s + (4.39 + 1.42i)29-s + (26.0 + 16.7i)31-s + ⋯ |
| L(s) = 1 | + (1.05 + 0.946i)3-s + (0.823 − 1.42i)5-s + (0.144 − 1.37i)7-s + (0.104 + 0.994i)9-s + (−0.0288 − 0.0646i)11-s + (−0.387 + 1.82i)13-s + (2.21 − 0.720i)15-s + (−0.595 + 1.33i)17-s + (−1.10 + 0.234i)19-s + (1.45 − 1.31i)21-s + (−0.0668 − 0.0920i)23-s + (−0.857 − 1.48i)25-s + (0.000368 − 0.000507i)27-s + (0.151 + 0.0492i)29-s + (0.841 + 0.540i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.04811 - 0.0280344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04811 - 0.0280344i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-26.0 - 16.7i)T \) |
| good | 3 | \( 1 + (-3.15 - 2.83i)T + (0.940 + 8.95i)T^{2} \) |
| 5 | \( 1 + (-4.11 + 7.13i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.01 + 9.65i)T + (-47.9 - 10.1i)T^{2} \) |
| 11 | \( 1 + (0.316 + 0.711i)T + (-80.9 + 89.9i)T^{2} \) |
| 13 | \( 1 + (5.04 - 23.7i)T + (-154. - 68.7i)T^{2} \) |
| 17 | \( 1 + (10.1 - 22.7i)T + (-193. - 214. i)T^{2} \) |
| 19 | \( 1 + (20.9 - 4.45i)T + (329. - 146. i)T^{2} \) |
| 23 | \( 1 + (1.53 + 2.11i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-4.39 - 1.42i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (13.7 - 7.93i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-41.7 - 46.3i)T + (-175. + 1.67e3i)T^{2} \) |
| 43 | \( 1 + (10.1 + 47.9i)T + (-1.68e3 + 752. i)T^{2} \) |
| 47 | \( 1 + (1.29 + 3.97i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-22.0 + 2.32i)T + (2.74e3 - 584. i)T^{2} \) |
| 59 | \( 1 + (29.6 - 32.9i)T + (-363. - 3.46e3i)T^{2} \) |
| 61 | \( 1 + 59.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-21.3 + 36.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (5.40 + 51.3i)T + (-4.93e3 + 1.04e3i)T^{2} \) |
| 73 | \( 1 + (7.22 + 16.2i)T + (-3.56e3 + 3.96e3i)T^{2} \) |
| 79 | \( 1 + (-48.2 + 108. i)T + (-4.17e3 - 4.63e3i)T^{2} \) |
| 83 | \( 1 + (64.4 - 58.0i)T + (720. - 6.85e3i)T^{2} \) |
| 89 | \( 1 + (-2.57 + 3.54i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (126. + 92.2i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.51749646044084198148746296694, −12.38014110341870012429048581789, −10.71051421960757547910409643725, −9.829108642411585186430795111537, −8.973334063042516969559157464142, −8.258337724325695511264310383741, −6.51449931410356865750105310393, −4.57274089907893182880917683880, −4.12057026114593299109008636874, −1.79816252320915602180915149723,
2.49459102584771507606740714079, 2.71028017939866346204866109585, 5.56315397417676212874267510411, 6.69226909479148984837927405559, 7.74154798139106728751207796850, 8.806226634644942702601498518415, 9.922751383797431209100813899840, 11.09235050398728313118535674446, 12.44026716934808906660762121251, 13.28497938353833577678506573361
