L(s) = 1 | + 3.19i·3-s + 5-s + (2.59 − 0.525i)7-s − 7.23·9-s + 3.34·11-s + 3.90·13-s + 3.19i·15-s − 2.92i·17-s + 6.33i·19-s + (1.68 + 8.29i)21-s + 3.44i·23-s + 25-s − 13.5i·27-s + 2.68i·29-s + 2.52·31-s + ⋯ |
L(s) = 1 | + 1.84i·3-s + 0.447·5-s + (0.980 − 0.198i)7-s − 2.41·9-s + 1.00·11-s + 1.08·13-s + 0.826i·15-s − 0.710i·17-s + 1.45i·19-s + (0.366 + 1.81i)21-s + 0.718i·23-s + 0.200·25-s − 2.61i·27-s + 0.497i·29-s + 0.453·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.040765845\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.040765845\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.59 + 0.525i)T \) |
good | 3 | \( 1 - 3.19iT - 3T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 3.90T + 13T^{2} \) |
| 17 | \( 1 + 2.92iT - 17T^{2} \) |
| 19 | \( 1 - 6.33iT - 19T^{2} \) |
| 23 | \( 1 - 3.44iT - 23T^{2} \) |
| 29 | \( 1 - 2.68iT - 29T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 - 4.70iT - 37T^{2} \) |
| 41 | \( 1 + 5.59iT - 41T^{2} \) |
| 43 | \( 1 + 8.62T + 43T^{2} \) |
| 47 | \( 1 - 0.506T + 47T^{2} \) |
| 53 | \( 1 + 11.4iT - 53T^{2} \) |
| 59 | \( 1 + 0.802iT - 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 - 1.11iT - 71T^{2} \) |
| 73 | \( 1 + 5.91iT - 73T^{2} \) |
| 79 | \( 1 + 10.5iT - 79T^{2} \) |
| 83 | \( 1 - 4.29iT - 83T^{2} \) |
| 89 | \( 1 + 2.00iT - 89T^{2} \) |
| 97 | \( 1 - 6.06iT - 97T^{2} \) |
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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
−10.15704875139814901539492492007, −9.316741808048547752362352662101, −8.723565274556107539061743545944, −7.945066813356735996803084220458, −6.48119147758880497828185622636, −5.57279283913996351831672303254, −4.87049461343520043699650579705, −3.95190413372822636407151323248, −3.31102451629605035860800932666, −1.58500862881503983071344195183,
1.04183101989741240900579291263, 1.79653246344552126091756004930, 2.84414173718037702448099003207, 4.39961171340731282755769732759, 5.67894687628465972207337628389, 6.35531815090584674825849497741, 6.95719504039669181723142928993, 7.959155993052166818189819239590, 8.578162834950648889085823245040, 9.173468364114977577579634496425