| L(s) = 1 | + (−0.831 − 1.14i)2-s + (−0.618 + 1.90i)4-s + (−1.74 − 1.26i)5-s + (−1.27 − 0.414i)7-s + (2.68 − 0.874i)8-s + 3.05i·10-s + (−10.9 + 0.273i)11-s + (−6.36 − 8.75i)13-s + (0.586 + 1.80i)14-s + (−3.23 − 2.35i)16-s + (−15.8 + 21.8i)17-s + (−9.95 + 3.23i)19-s + (3.49 − 2.53i)20-s + (9.45 + 12.3i)22-s − 14.1·23-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (−0.154 + 0.475i)4-s + (−0.349 − 0.253i)5-s + (−0.182 − 0.0592i)7-s + (0.336 − 0.109i)8-s + 0.305i·10-s + (−0.999 + 0.0248i)11-s + (−0.489 − 0.673i)13-s + (0.0418 + 0.128i)14-s + (−0.202 − 0.146i)16-s + (−0.934 + 1.28i)17-s + (−0.523 + 0.170i)19-s + (0.174 − 0.126i)20-s + (0.429 + 0.561i)22-s − 0.617·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0224984 + 0.0822640i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0224984 + 0.0822640i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.831 + 1.14i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (10.9 - 0.273i)T \) |
| good | 5 | \( 1 + (1.74 + 1.26i)T + (7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (1.27 + 0.414i)T + (39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (6.36 + 8.75i)T + (-52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (15.8 - 21.8i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (9.95 - 3.23i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + 14.1T + 529T^{2} \) |
| 29 | \( 1 + (-11.7 - 3.81i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (20.7 - 15.0i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (9.00 - 27.7i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-10.1 + 3.30i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 - 41.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (21.6 + 66.4i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-48.5 + 35.2i)T + (868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-16.2 + 49.8i)T + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-64.7 + 89.1i)T + (-1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 - 60.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-56.0 - 40.7i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (69.7 + 22.6i)T + (4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (58.9 + 81.1i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (59.8 - 82.4i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 - 55.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-141. + 102. i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−11.62853211877084825356454011775, −10.54307995021511710217260138591, −9.980958667835607625218479201437, −8.505858350924790001752493698639, −7.999629917276912053528899298589, −6.56400685990676342303586384545, −5.03887981668602950527777509144, −3.72478339363313436002620278025, −2.20465576774539150891464820502, −0.05203291511395295521482444695,
2.47217721292571897493449970407, 4.34274325276307722990796546109, 5.57596774903246388509429806941, 6.91031608547986572849487859195, 7.61265847195073226198235174010, 8.828332738471788005711081254921, 9.708078930955208192115263728895, 10.81396235535089496536696199677, 11.67698090644944806135097462338, 12.94135817372651288907365832189
