L(s) = 1 | + (−0.366 − 0.633i)2-s + (−0.5 + 0.866i)3-s + (0.732 − 1.26i)4-s + (0.5 + 0.866i)5-s + 0.732·6-s − 2.53·8-s + (−0.499 − 0.866i)9-s + (0.366 − 0.633i)10-s + (1.36 − 2.36i)11-s + (0.732 + 1.26i)12-s − 5.73·13-s − 0.999·15-s + (−0.535 − 0.928i)16-s + (3.36 − 5.83i)17-s + (−0.366 + 0.633i)18-s + (−1.23 − 2.13i)19-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.448i)2-s + (−0.288 + 0.499i)3-s + (0.366 − 0.633i)4-s + (0.223 + 0.387i)5-s + 0.298·6-s − 0.896·8-s + (−0.166 − 0.288i)9-s + (0.115 − 0.200i)10-s + (0.411 − 0.713i)11-s + (0.211 + 0.366i)12-s − 1.58·13-s − 0.258·15-s + (−0.133 − 0.232i)16-s + (0.816 − 1.41i)17-s + (−0.0862 + 0.149i)18-s + (−0.282 − 0.489i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.561066 - 0.843478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.561066 - 0.843478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.366 + 0.633i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 + (-3.36 + 5.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.23 + 2.13i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.633 - 1.09i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + (-3.23 + 5.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.59 + 6.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 7.19T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.19 + 7.26i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.09 + 8.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.33 - 2.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + (2.33 - 4.03i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.69 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + (4.56 + 7.90i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.07T + 97T^{2} \) |
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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
−9.962853813876190852916624158394, −9.708973876764694101394155476029, −8.721372250765560766100448429187, −7.33341514600720588247743350159, −6.56811752403602120406425146070, −5.54465980339993046858210262925, −4.81589033667382310491896729310, −3.23320708775288127044705645235, −2.35302032581317506408621592450, −0.57055267335360952873773991915,
1.67477989739507423860542562299, 2.93093677127986404445460401394, 4.35431459642164100505704132356, 5.45047405511380910599690770327, 6.53567334321673434683030051952, 7.08805484137193944006437138104, 8.074919716569907113582633073196, 8.603923834807982323163367721286, 9.840579074815390818761906790068, 10.44668475019042233409137032774