L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 5-s + (−1 − 1.73i)7-s + 0.999·8-s + (−0.5 + 0.866i)10-s + (−1.5 + 2.59i)11-s + (1 − 3.46i)13-s + 1.99·14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (−1 − 1.73i)19-s + (−0.499 − 0.866i)20-s + (−1.5 − 2.59i)22-s + (1.5 − 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (−0.158 + 0.273i)10-s + (−0.452 + 0.783i)11-s + (0.277 − 0.960i)13-s + 0.534·14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (−0.229 − 0.397i)19-s + (−0.111 − 0.193i)20-s + (−0.319 − 0.553i)22-s + (0.312 − 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.326633520\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.326633520\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6 + 10.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.898045651769643600425840887257, −8.891070491682525532386658414183, −8.027689691280649030350978385673, −7.38393643800771498406094090820, −6.43301638102232684230158465435, −5.74666410942358355966111509415, −4.75035014705503777374577663516, −3.69954522738404830254584846954, −2.32901142162861317005420996765, −0.791444958194323520660811044946,
1.13259063017847461427288780275, 2.53773791588137724588296101243, 3.22900913174491688577637999713, 4.55150920627240293930766179360, 5.56679264005520412935511674874, 6.37696470863339451746065675156, 7.43220661310585432294035941412, 8.396685719652457620059457196034, 9.131266351065574372980335278591, 9.702932306045277440553716292731