| L(s) = 1 | + (−0.862 + 1.12i)2-s + (−0.510 − 1.93i)4-s − i·5-s − i·7-s + (2.60 + 1.09i)8-s + (1.12 + 0.862i)10-s − 4.23·11-s − 1.31·13-s + (1.12 + 0.862i)14-s + (−3.47 + 1.97i)16-s + 3.77i·17-s − 4.64i·19-s + (−1.93 + 0.510i)20-s + (3.65 − 4.74i)22-s − 1.08·23-s + ⋯ |
| L(s) = 1 | + (−0.610 + 0.792i)2-s + (−0.255 − 0.966i)4-s − 0.447i·5-s − 0.377i·7-s + (0.921 + 0.387i)8-s + (0.354 + 0.272i)10-s − 1.27·11-s − 0.364·13-s + (0.299 + 0.230i)14-s + (−0.869 + 0.493i)16-s + 0.914i·17-s − 1.06i·19-s + (−0.432 + 0.114i)20-s + (0.779 − 1.01i)22-s − 0.226·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3558438763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3558438763\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.862 - 1.12i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 - 3.77iT - 17T^{2} \) |
| 19 | \( 1 + 4.64iT - 19T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 - 7.76iT - 29T^{2} \) |
| 31 | \( 1 + 2.35iT - 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 - 11.6iT - 41T^{2} \) |
| 43 | \( 1 - 11.9iT - 43T^{2} \) |
| 47 | \( 1 - 1.69T + 47T^{2} \) |
| 53 | \( 1 - 2.07iT - 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 5.42iT - 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 + 1.47T + 73T^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + 3.23T + 83T^{2} \) |
| 89 | \( 1 - 18.6iT - 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−9.848536866456495747297028213543, −9.209998347486665444791764494999, −8.207522543443638760772911506332, −7.79611024292102738338960457450, −6.86217930905656882872742357663, −5.98199477417488563758829845872, −5.04916730452214876303323775625, −4.42065543671817569039960719906, −2.78550685808957542080429613671, −1.33197173678506128980554890423,
0.18751300139326181546195649778, 2.05789792113939225087457814365, 2.78037891482430817977748116022, 3.82709372188924712607109064541, 4.97580607705716239693614577366, 5.93823148361355993162894928522, 7.27441955528997643726948227556, 7.72787297860494262016978964523, 8.627882666793119934299039011836, 9.453103212378992214135297847895
