L(s) = 1 | + (−0.556 + 5.16i)3-s − 10.6i·5-s + 7.90i·7-s + (−26.3 − 5.75i)9-s − 11.7·11-s + 30.5·13-s + (54.8 + 5.90i)15-s − 118. i·17-s + 66.4i·19-s + (−40.8 − 4.40i)21-s + 166.·23-s + 12.4·25-s + (44.4 − 133. i)27-s + 111. i·29-s − 224. i·31-s + ⋯ |
L(s) = 1 | + (−0.107 + 0.994i)3-s − 0.948i·5-s + 0.426i·7-s + (−0.977 − 0.213i)9-s − 0.322·11-s + 0.652·13-s + (0.943 + 0.101i)15-s − 1.68i·17-s + 0.802i·19-s + (−0.424 − 0.0457i)21-s + 1.50·23-s + 0.0998·25-s + (0.316 − 0.948i)27-s + 0.717i·29-s − 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.682490419\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.682490419\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.556 - 5.16i)T \) |
good | 5 | \( 1 + 10.6iT - 125T^{2} \) |
| 7 | \( 1 - 7.90iT - 343T^{2} \) |
| 11 | \( 1 + 11.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 66.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 111. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 224. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 70.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 247. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 98.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 529. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 811.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 833.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 2.83iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 796.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 875.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 656. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 237.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 868. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 755.T + 9.12e5T^{2} \) |
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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
−10.91859158716324639953059635926, −9.871135305564044439025100771291, −9.012939591182058960614459385071, −8.548927216130733219658245761967, −7.16337332166446029992021531288, −5.62539700103097022849938327932, −5.11743065434242594491908658085, −4.00061979356112389245580588693, −2.71221628916112011594792371716, −0.73282755587719211496966884912,
1.06251116560897570402594862086, 2.50788419320440672646265099376, 3.62562212490150747512106440541, 5.28072888037850276477690702714, 6.51446879454661761518564127372, 6.93363235112178388113350095635, 8.044589762580535678165843489858, 8.844843229864734426616597665350, 10.39040633898077679564495510395, 10.90746821536245493019613516834