| L(s) = 1 | − i·3-s + (−1.63 + 1.52i)5-s − 3.48i·7-s − 9-s − 1.37·11-s − 0.00615i·13-s + (1.52 + 1.63i)15-s − 0.950i·17-s − 5.93·19-s − 3.48·21-s + 5.79i·23-s + (0.346 − 4.98i)25-s + i·27-s + 1.84·29-s − 4.21·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.731 + 0.682i)5-s − 1.31i·7-s − 0.333·9-s − 0.415·11-s − 0.00170i·13-s + (0.393 + 0.422i)15-s − 0.230i·17-s − 1.36·19-s − 0.760·21-s + 1.20i·23-s + (0.0692 − 0.997i)25-s + 0.192i·27-s + 0.342·29-s − 0.756·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8778674073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8778674073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.63 - 1.52i)T \) |
| 67 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.48iT - 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 0.00615iT - 13T^{2} \) |
| 17 | \( 1 + 0.950iT - 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 - 5.79iT - 23T^{2} \) |
| 29 | \( 1 - 1.84T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 8.70iT - 37T^{2} \) |
| 41 | \( 1 - 5.62T + 41T^{2} \) |
| 43 | \( 1 + 1.88iT - 43T^{2} \) |
| 47 | \( 1 - 5.91iT - 47T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 8.73T + 61T^{2} \) |
| 71 | \( 1 - 5.85T + 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 7.34T + 79T^{2} \) |
| 83 | \( 1 + 17.1iT - 83T^{2} \) |
| 89 | \( 1 + 0.0825T + 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−8.104169424206567866001963115366, −7.87542923842944290398782025160, −6.93545015562184102585096357376, −6.74387580571392695229523777083, −5.67274746284209155446335014468, −4.60467654893401226213797408296, −3.88061786871794095433877604928, −3.14474846788353675190919227030, −2.09537749111853684614393083584, −0.834824958272276714814338263889,
0.32726513798775656047542521933, 2.02977172242708534248208046063, 2.81081785606836811905756269576, 3.94820950076199441908693420274, 4.50869319101179063018883216854, 5.37073229917319511711841945831, 5.91316542770238331653731255547, 6.86492286656372538307972606818, 7.934727586794848768095732286718, 8.446782957539992261279030433944
