| L(s) = 1 | + (−0.170 + 1.62i)2-s + (1.63 + 0.585i)3-s + (−0.647 − 0.137i)4-s − 4.38·5-s + (−1.22 + 2.54i)6-s + (0.293 + 0.903i)7-s + (−0.674 + 2.07i)8-s + (2.31 + 1.90i)9-s + (0.748 − 7.12i)10-s + (−0.605 − 0.128i)11-s + (−0.975 − 0.603i)12-s + (−3.39 + 2.46i)13-s + (−1.51 + 0.322i)14-s + (−7.15 − 2.57i)15-s + (−4.46 − 1.98i)16-s + (3.33 + 3.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.120 + 1.14i)2-s + (0.941 + 0.338i)3-s + (−0.323 − 0.0688i)4-s − 1.96·5-s + (−0.501 + 1.03i)6-s + (0.110 + 0.341i)7-s + (−0.238 + 0.733i)8-s + (0.771 + 0.636i)9-s + (0.236 − 2.25i)10-s + (−0.182 − 0.0388i)11-s + (−0.281 − 0.174i)12-s + (−0.940 + 0.683i)13-s + (−0.405 + 0.0860i)14-s + (−1.84 − 0.663i)15-s + (−1.11 − 0.496i)16-s + (0.809 + 0.899i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.176615 + 1.15731i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.176615 + 1.15731i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.63 - 0.585i)T \) |
| 31 | \( 1 + (-5.53 + 0.601i)T \) |
| good | 2 | \( 1 + (0.170 - 1.62i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + 4.38T + 5T^{2} \) |
| 7 | \( 1 + (-0.293 - 0.903i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.605 + 0.128i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.39 - 2.46i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.33 - 3.70i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.07 + 1.81i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.376 - 0.418i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.800 + 7.61i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (2.22 - 3.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.65 - 2.65i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (4.88 + 3.55i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (5.01 + 2.23i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-5.13 + 1.09i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-4.83 - 2.15i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-4.37 - 7.58i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + (1.47 - 0.313i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-2.31 + 2.56i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (0.502 - 1.54i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.76 + 1.23i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.37 - 4.23i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (10.4 - 11.5i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−11.98790894064620983514621062143, −11.67200929005554601786967638810, −10.22773384211590360567781651964, −8.941456703364890428237866218190, −8.121127816826912475800784221985, −7.66825943384481297328176089920, −6.83619962521327964168364070652, −5.09689230923747560614541221542, −4.09392030994276016126016606100, −2.80930581903628409030699307648,
0.872694266943845252681652934085, 2.92427044525458747158712076866, 3.48639721515514399380048581930, 4.70574482364594753542944781321, 7.15437125112745117602145825518, 7.52915473938511470174727639906, 8.543784303045260096170000966848, 9.717764888403991268724994265200, 10.58725653086342683038987825558, 11.65346318725104417554918446029
