L(s) = 1 | + (0.847 + 1.13i)2-s + (−0.562 + 1.91i)4-s − 2.55i·5-s + 3.08i·7-s + (−2.64 + 0.990i)8-s + (2.88 − 2.16i)10-s − 5.76·11-s − 1.95·13-s + (−3.49 + 2.61i)14-s + (−3.36 − 2.15i)16-s − 1.39·17-s + (3.64 − 2.39i)19-s + (4.89 + 1.43i)20-s + (−4.88 − 6.52i)22-s − 1.59i·23-s + ⋯ |
L(s) = 1 | + (0.599 + 0.800i)2-s + (−0.281 + 0.959i)4-s − 1.14i·5-s + 1.16i·7-s + (−0.936 + 0.350i)8-s + (0.913 − 0.684i)10-s − 1.73·11-s − 0.541·13-s + (−0.934 + 0.700i)14-s + (−0.841 − 0.539i)16-s − 0.339·17-s + (0.835 − 0.550i)19-s + (1.09 + 0.321i)20-s + (−1.04 − 1.39i)22-s − 0.331i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1559208302\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1559208302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.847 - 1.13i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.64 + 2.39i)T \) |
good | 5 | \( 1 + 2.55iT - 5T^{2} \) |
| 7 | \( 1 - 3.08iT - 7T^{2} \) |
| 11 | \( 1 + 5.76T + 11T^{2} \) |
| 13 | \( 1 + 1.95T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 23 | \( 1 + 1.59iT - 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 6.86iT - 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 - 8.93iT - 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 12.8iT - 59T^{2} \) |
| 61 | \( 1 - 6.56iT - 61T^{2} \) |
| 67 | \( 1 - 16.2iT - 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 - 3.48T + 73T^{2} \) |
| 79 | \( 1 + 8.99T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 - 9.05iT - 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.801545666148958990199290519601, −9.021845068033653551151648936474, −8.420718718277107929199076366798, −7.76009105212693158136312385264, −6.83977604941154912713115345698, −5.56662087060548146862044563301, −5.26551662909470851361143197856, −4.62443147664603114833020217666, −3.17579919347012577576020895171, −2.24119617507975160202380088586,
0.04719634976512261146579109080, 1.85104885391066010480126720634, 2.99692518041567778920696182689, 3.55161642218465756737306199999, 4.77363675954085997100715615147, 5.47640816976470748824104656364, 6.62932101345636673969891792909, 7.32433074613421306401631271673, 8.096191404453968578257950556206, 9.538409171935392084617586772501