L(s) = 1 | + (0.5 − 0.866i)3-s + (2.78 − 1.60i)5-s + (−0.499 − 0.866i)9-s + (1.18 + 0.683i)11-s − 2.93i·13-s − 3.21i·15-s + (−5.98 − 3.45i)17-s + (−3.67 − 6.36i)19-s + (−3.14 + 1.81i)23-s + (2.66 − 4.62i)25-s − 0.999·27-s − 1.11·29-s + (−4.35 + 7.53i)31-s + (1.18 − 0.683i)33-s + (−3.81 − 6.61i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (1.24 − 0.718i)5-s + (−0.166 − 0.288i)9-s + (0.356 + 0.206i)11-s − 0.812i·13-s − 0.830i·15-s + (−1.45 − 0.838i)17-s + (−0.843 − 1.46i)19-s + (−0.655 + 0.378i)23-s + (0.533 − 0.924i)25-s − 0.192·27-s − 0.206·29-s + (−0.781 + 1.35i)31-s + (0.206 − 0.118i)33-s + (−0.627 − 1.08i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.936484087\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.936484087\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.78 + 1.60i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.18 - 0.683i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 + (5.98 + 3.45i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.67 + 6.36i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.14 - 1.81i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 + (4.35 - 7.53i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.81 + 6.61i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.833iT - 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 + (1.47 + 2.55i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.28 + 3.96i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.04 + 12.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.57 + 5.53i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.5 - 6.07i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (-5.62 - 3.24i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.74 + 3.89i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 + (-10.9 + 6.30i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.893758324360009076844797030825, −8.089805172108354506593032090818, −6.87666386761339835529469046133, −6.62064911991613603186661241042, −5.41009936659322558601801973588, −4.99421918592379841826957603976, −3.78857503966998915166416950670, −2.46118163830090786301185223897, −1.91292312349485684265856584721, −0.56742443791190630739745746497,
1.90567000229695492126029684291, 2.27045567211827485949823530820, 3.70256229899562092186933323684, 4.24808858024350197223735035283, 5.47448419692702907703251218988, 6.32553022017595400660843232690, 6.57424175378969174016170243814, 7.83777936220809090418273749153, 8.703477385390687224673179110653, 9.259742283533207568680972656041