L(s) = 1 | + (1.01 − 1.40i)3-s + (1.60 + 2.77i)5-s + (−0.555 + 2.58i)7-s + (−0.927 − 2.85i)9-s + (4.13 + 2.38i)11-s + (−0.861 + 0.497i)13-s + (5.52 + 0.580i)15-s − 6.51·17-s + 5.38i·19-s + (3.05 + 3.41i)21-s + (6.56 − 3.79i)23-s + (−2.63 + 4.57i)25-s + (−4.94 − 1.60i)27-s + (−2.93 − 1.69i)29-s + (3.51 − 2.02i)31-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)3-s + (0.716 + 1.24i)5-s + (−0.209 + 0.977i)7-s + (−0.309 − 0.951i)9-s + (1.24 + 0.720i)11-s + (−0.239 + 0.137i)13-s + (1.42 + 0.149i)15-s − 1.57·17-s + 1.23i·19-s + (0.667 + 0.744i)21-s + (1.36 − 0.790i)23-s + (−0.527 + 0.914i)25-s + (−0.951 − 0.309i)27-s + (−0.545 − 0.314i)29-s + (0.630 − 0.364i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85182 + 0.425652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85182 + 0.425652i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.01 + 1.40i)T \) |
| 7 | \( 1 + (0.555 - 2.58i)T \) |
good | 5 | \( 1 + (-1.60 - 2.77i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.13 - 2.38i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.861 - 0.497i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6.51T + 17T^{2} \) |
| 19 | \( 1 - 5.38iT - 19T^{2} \) |
| 23 | \( 1 + (-6.56 + 3.79i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.93 + 1.69i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.51 + 2.02i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + (3.48 + 6.04i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.81 + 6.60i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.78 + 6.54i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 7.77iT - 53T^{2} \) |
| 59 | \( 1 + (2.25 + 3.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.26 - 3.03i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.493 - 0.854i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.20iT - 71T^{2} \) |
| 73 | \( 1 - 2.63iT - 73T^{2} \) |
| 79 | \( 1 + (7.96 - 13.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.49 - 7.77i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 4.42T + 89T^{2} \) |
| 97 | \( 1 + (3.13 + 1.81i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.08228483301902603979353466821, −9.883544631444448171785570763990, −9.207014415062353741301279149991, −8.407038869064414255045641603162, −6.90197070671309921523868785393, −6.73816233897139067945777708146, −5.73844527132703341555026011941, −3.96634144571331134381742270351, −2.61684074806976894655905754485, −1.97485982188257096575278749234,
1.20659487549192158012377797047, 2.95517615765208082552413146706, 4.31195271328738952059973174002, 4.81986207845026345978086954554, 6.14302284422400937896530775529, 7.27040697458926820291432257263, 8.595209018714241658166006727207, 9.176906716333042510980332667133, 9.601101962901670066454746022126, 10.88582905562644127810710827602