| L(s) = 1 | + (−1.41 − 1.73i)5-s − 2.44·7-s + 1.41i·11-s + 2.44·13-s − 3.46·17-s − 6·19-s − 6.92i·23-s + (−0.999 + 4.89i)25-s + 5.65·29-s + 2i·31-s + (3.46 + 4.24i)35-s + 7.34·37-s + 4.24i·41-s + 4.89i·43-s + 3.46i·47-s + ⋯ |
| L(s) = 1 | + (−0.632 − 0.774i)5-s − 0.925·7-s + 0.426i·11-s + 0.679·13-s − 0.840·17-s − 1.37·19-s − 1.44i·23-s + (−0.199 + 0.979i)25-s + 1.05·29-s + 0.359i·31-s + (0.585 + 0.717i)35-s + 1.20·37-s + 0.662i·41-s + 0.747i·43-s + 0.505i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8952883775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8952883775\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
| good | 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 - 7.34T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 1.41iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 - 14.6iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−8.624105212151154344250568565241, −8.448366477839832300151310529292, −7.35547125808631386482755403834, −6.46742445832680543709911380578, −6.07300110407005410949380507799, −4.56923570385183463160321072098, −4.44098943915066058792052526604, −3.29114422296748561612133504504, −2.27668675301760945203107859196, −0.855864800327848762963349556569,
0.37597171339505609728724607759, 2.10128598704338118334297723476, 3.12963567901789179942801515866, 3.75829877363639456715144667288, 4.57169705565015117731327931765, 5.91605842395752422602950614197, 6.39439200488153005219779538647, 7.04938230490423221240727466829, 7.941202199719078565023461579145, 8.621165755142995124273084229292
