| L(s) = 1 | + (−0.781 + 1.54i)3-s + 1.94i·5-s + 3.97i·7-s + (−1.77 − 2.41i)9-s − 5.57·11-s − 1.25·13-s + (−3.00 − 1.51i)15-s − i·17-s − 2.07i·19-s + (−6.13 − 3.10i)21-s + 4.52·23-s + 1.22·25-s + (5.12 − 0.861i)27-s − 7.39i·29-s + 8.94i·31-s + ⋯ |
| L(s) = 1 | + (−0.451 + 0.892i)3-s + 0.868i·5-s + 1.50i·7-s + (−0.592 − 0.805i)9-s − 1.68·11-s − 0.347·13-s + (−0.775 − 0.391i)15-s − 0.242i·17-s − 0.476i·19-s + (−1.33 − 0.677i)21-s + 0.942·23-s + 0.245·25-s + (0.986 − 0.165i)27-s − 1.37i·29-s + 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.163238 - 0.547897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.163238 - 0.547897i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.781 - 1.54i)T \) |
| 17 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.94iT - 5T^{2} \) |
| 7 | \( 1 - 3.97iT - 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 19 | \( 1 + 2.07iT - 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 + 7.39iT - 29T^{2} \) |
| 31 | \( 1 - 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 1.33T + 37T^{2} \) |
| 41 | \( 1 - 0.747iT - 41T^{2} \) |
| 43 | \( 1 + 4.26iT - 43T^{2} \) |
| 47 | \( 1 + 4.99T + 47T^{2} \) |
| 53 | \( 1 - 1.80iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 + 0.151T + 71T^{2} \) |
| 73 | \( 1 + 1.15T + 73T^{2} \) |
| 79 | \( 1 + 3.97iT - 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 17.9iT - 89T^{2} \) |
| 97 | \( 1 + 9.97T + 97T^{2} \) |
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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
−10.70684626982784804613108716733, −10.03161644523068365861552316829, −9.144625253644791131302670783413, −8.383329061909244576242653843328, −7.23490835795550051455117839047, −6.19004996715686340001028411778, −5.35643286747925451856447648331, −4.74921346721112019500782843606, −3.03664202538270229799807820819, −2.61947654591466063050910776735,
0.29522769987639056554866472447, 1.50226198691380108274146900908, 3.02059833878716959669040724624, 4.57501158360771434342608278359, 5.18517564775566755612169604566, 6.27672936860880598273796301054, 7.44395396485919816261462591057, 7.70654351895453779306184861151, 8.682655836286810371666173194971, 9.920831843202922134694678168950
