L(s) = 1 | − 2.23i·5-s + 4.47i·17-s − 4·19-s + 8.94i·23-s − 5.00·25-s − 8·31-s + 8.94i·47-s + 7·49-s − 4.47i·53-s − 2·61-s + 16·79-s + 17.8i·83-s + 10.0·85-s + 8.94i·95-s + 17.8i·107-s + ⋯ |
L(s) = 1 | − 0.999i·5-s + 1.08i·17-s − 0.917·19-s + 1.86i·23-s − 1.00·25-s − 1.43·31-s + 1.30i·47-s + 49-s − 0.614i·53-s − 0.256·61-s + 1.80·79-s + 1.96i·83-s + 1.08·85-s + 0.917i·95-s + 1.72i·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9383282076\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9383282076\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.23iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8.94iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8.94iT - 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 17.8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.091482986681404562971889931575, −8.148727247112828420893624694951, −7.68394655962824975222945833267, −6.63561446340073798517536420845, −5.75174749472223617614871891128, −5.20771070721478966217519539551, −4.15810690148476979761528942095, −3.59212897929211924763028621500, −2.13069020880937209578271275115, −1.27023321969621178868835315947,
0.29828260348684739435647604936, 2.06047849441444224348186328196, 2.77471491444084888192010752235, 3.75680133844636111863684049269, 4.61262173774234852615432075539, 5.59304205312024043906668414626, 6.45522025376838678951888075438, 7.00574336656910988627111994058, 7.71268986441791687435106126101, 8.638401598116661544990228385473