| L(s) = 1 | + (−0.540 − 0.841i)2-s + (1.29 − 1.14i)3-s + (−0.415 + 0.909i)4-s + (−1.43 + 0.422i)5-s + (−1.66 − 0.469i)6-s + (3.34 − 2.89i)7-s + (0.989 − 0.142i)8-s + (0.362 − 2.97i)9-s + (1.13 + 0.981i)10-s + (−1.75 − 1.12i)11-s + (0.505 + 1.65i)12-s + (−2.50 + 2.88i)13-s + (−4.24 − 1.24i)14-s + (−1.37 + 2.19i)15-s + (−0.654 − 0.755i)16-s + (1.53 + 3.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.594i)2-s + (0.748 − 0.662i)3-s + (−0.207 + 0.454i)4-s + (−0.642 + 0.188i)5-s + (−0.680 − 0.191i)6-s + (1.26 − 1.09i)7-s + (0.349 − 0.0503i)8-s + (0.120 − 0.992i)9-s + (0.358 + 0.310i)10-s + (−0.528 − 0.339i)11-s + (0.146 + 0.478i)12-s + (−0.694 + 0.801i)13-s + (−1.13 − 0.332i)14-s + (−0.356 + 0.567i)15-s + (−0.163 − 0.188i)16-s + (0.372 + 0.814i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0925 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0925 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.803619 - 0.732370i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.803619 - 0.732370i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 + 0.841i)T \) |
| 3 | \( 1 + (-1.29 + 1.14i)T \) |
| 23 | \( 1 + (-4.39 - 1.92i)T \) |
| good | 5 | \( 1 + (1.43 - 0.422i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-3.34 + 2.89i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (1.75 + 1.12i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.50 - 2.88i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 3.35i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.806 - 0.368i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-4.85 + 2.21i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.34 - 9.32i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.717 - 2.44i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.0613 + 0.209i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (3.01 + 0.433i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 9.62iT - 47T^{2} \) |
| 53 | \( 1 + (8.08 + 9.32i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (6.92 + 6.00i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-6.28 + 0.903i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.67 + 2.60i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-6.11 - 9.50i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.78 - 3.91i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (9.86 + 8.54i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (3.42 + 1.00i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.50 + 10.4i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.95 + 6.65i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−12.89801999090375819650612248082, −11.82242803204330526966398738715, −11.03545729415343322381052937497, −9.921064884983835113764471700419, −8.489115126370035706832609239861, −7.82036827654893150162757882168, −6.98363779097904201053117149378, −4.62171219906618617133786422214, −3.31072776382956083178520121905, −1.50151487639352558138632540121,
2.58353133674491013180686074384, 4.64424647556260584316977057366, 5.37069944700994717959544131447, 7.55685077501638919443061132169, 8.117936200696149474042521978083, 9.037206882198360919966223947437, 10.10997770120989258663194764494, 11.28148279906720346793889544250, 12.35788733961325267940718731983, 13.76881949428642008575506441875
