L(s) = 1 | + (0.847 + 1.13i)2-s + (−0.242 − 1.71i)3-s + (−0.562 + 1.91i)4-s + (1.73 − 1.72i)6-s − 3.08i·7-s + (−2.64 + 0.990i)8-s + (−2.88 + 0.831i)9-s − 2.54i·11-s + (3.42 + 0.499i)12-s − 5.06i·13-s + (3.49 − 2.61i)14-s + (−3.36 − 2.15i)16-s + 0.214i·17-s + (−3.38 − 2.55i)18-s + 2.60·19-s + ⋯ |
L(s) = 1 | + (0.599 + 0.800i)2-s + (−0.139 − 0.990i)3-s + (−0.281 + 0.959i)4-s + (0.708 − 0.705i)6-s − 1.16i·7-s + (−0.936 + 0.350i)8-s + (−0.960 + 0.277i)9-s − 0.767i·11-s + (0.989 + 0.144i)12-s − 1.40i·13-s + (0.934 − 0.700i)14-s + (−0.841 − 0.539i)16-s + 0.0519i·17-s + (−0.797 − 0.602i)18-s + 0.598·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30506 - 0.775719i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30506 - 0.775719i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.242 + 1.71i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.08iT - 7T^{2} \) |
| 11 | \( 1 + 2.54iT - 11T^{2} \) |
| 13 | \( 1 + 5.06iT - 13T^{2} \) |
| 17 | \( 1 - 0.214iT - 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 4.58iT - 31T^{2} \) |
| 37 | \( 1 + 7.67iT - 37T^{2} \) |
| 41 | \( 1 - 9.26iT - 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 9.51T + 53T^{2} \) |
| 59 | \( 1 - 0.428iT - 59T^{2} \) |
| 61 | \( 1 + 1.11iT - 61T^{2} \) |
| 67 | \( 1 - 2.35T + 67T^{2} \) |
| 71 | \( 1 - 6.12T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 2.29iT - 83T^{2} \) |
| 89 | \( 1 - 12.4iT - 89T^{2} \) |
| 97 | \( 1 + 8.04T + 97T^{2} \) |
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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
−10.86216157375667643821232588503, −9.436493917758570857914435383703, −8.271268747199265484477042379278, −7.63258548120353267802582342891, −7.04821398203290017160563779300, −5.94259531486015745515258857313, −5.31012971819292012449768357029, −3.85534747705879452245917803213, −2.85803360308196314463295259679, −0.71477343853539352064972526065,
1.97830526522241915480131701073, 3.12424410254573056397928870055, 4.25233243695726859836131277307, 5.08212431641315352063065050101, 5.83869843233780589338125594100, 6.99730667409536745745335211305, 8.822632416002893541528066508352, 9.213994757635179733941212751926, 9.964777195574544552640739223854, 10.95698552103954329710015226658