| L(s) = 1 | + (0.781 + 0.623i)2-s + (0.748 + 1.56i)3-s + (0.222 + 0.974i)4-s + (0.182 + 2.43i)5-s + (−0.389 + 1.68i)6-s + (0.231 − 2.63i)7-s + (−0.433 + 0.900i)8-s + (−1.88 + 2.33i)9-s + (−1.37 + 2.01i)10-s + (1.06 − 0.418i)11-s + (−1.35 + 1.07i)12-s + (−4.91 + 1.92i)13-s + (1.82 − 1.91i)14-s + (−3.66 + 2.10i)15-s + (−0.900 + 0.433i)16-s + (−4.82 + 1.48i)17-s + ⋯ |
| L(s) = 1 | + (0.552 + 0.440i)2-s + (0.431 + 0.901i)3-s + (0.111 + 0.487i)4-s + (0.0815 + 1.08i)5-s + (−0.158 + 0.689i)6-s + (0.0873 − 0.996i)7-s + (−0.153 + 0.318i)8-s + (−0.626 + 0.779i)9-s + (−0.434 + 0.637i)10-s + (0.321 − 0.126i)11-s + (−0.391 + 0.310i)12-s + (−1.36 + 0.534i)13-s + (0.487 − 0.512i)14-s + (−0.946 + 0.543i)15-s + (−0.225 + 0.108i)16-s + (−1.17 + 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.459346 + 2.09288i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.459346 + 2.09288i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 3 | \( 1 + (-0.748 - 1.56i)T \) |
| 7 | \( 1 + (-0.231 + 2.63i)T \) |
| good | 5 | \( 1 + (-0.182 - 2.43i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 0.418i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (4.91 - 1.92i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (4.82 - 1.48i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.26 - 1.30i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.41 - 5.83i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.909 - 2.94i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + 4.80iT - 31T^{2} \) |
| 37 | \( 1 + (-0.357 - 0.332i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-7.09 + 4.83i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-4.23 - 2.88i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (-4.94 + 6.19i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-5.77 - 6.22i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (6.20 - 2.98i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (11.6 + 2.64i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 0.738T + 67T^{2} \) |
| 71 | \( 1 + (-6.20 + 1.41i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-6.38 - 2.50i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 - 3.52T + 79T^{2} \) |
| 83 | \( 1 + (-3.13 + 8.00i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-13.5 + 2.04i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-5.16 + 2.98i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.66203291398661616665085127332, −9.617217086698889669662221419756, −8.982045618049993844905969697791, −7.55218182981171050346789292775, −7.25977221357258747470518078506, −6.21315856497320536022832213626, −5.01462919258095782784463838292, −4.20459526650217666879653066866, −3.36943571258376024888672481974, −2.35840648212811723522255221269,
0.803601918118143576126653570348, 2.23755909219740542759894616809, 2.88548565850361195528701023408, 4.58481862426601423669844531944, 5.16993748317562889758528324943, 6.24067584918844325247833399771, 7.15286882643786041056849838199, 8.198269399234500825297711756541, 9.133129707598455757474716030890, 9.352830389804553477111586873833
