| L(s) = 1 | + 5-s + 3.61·7-s + 2.34·11-s − 3.95·13-s + 2.34·17-s + 1.61·19-s − 2.61·23-s + 25-s + 4.61·29-s − 0.340·31-s + 3.61·35-s + 7.61·37-s − 9.90·41-s + 9.29·43-s − 1.38·47-s + 6.06·49-s + 4·53-s + 2.34·55-s + 11.2·59-s + 7.61·61-s − 3.95·65-s − 8.29·67-s − 13.9·71-s + 14.8·73-s + 8.45·77-s + 0.0452·79-s − 15.1·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.36·7-s + 0.705·11-s − 1.09·13-s + 0.567·17-s + 0.370·19-s − 0.545·23-s + 0.200·25-s + 0.856·29-s − 0.0610·31-s + 0.610·35-s + 1.25·37-s − 1.54·41-s + 1.41·43-s − 0.202·47-s + 0.866·49-s + 0.549·53-s + 0.315·55-s + 1.46·59-s + 0.974·61-s − 0.490·65-s − 1.01·67-s − 1.65·71-s + 1.73·73-s + 0.963·77-s + 0.00509·79-s − 1.66·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.671515073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.671515073\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 3.95T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 + 0.340T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 - 9.29T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 7.61T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 0.0452T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 4.38T + 97T^{2} \) |
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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
−8.354711609730548465701000111361, −7.65615425096118298080508648193, −7.04623862412338113697420853964, −6.10880545889739533497801635641, −5.33291685403171159343343644159, −4.72198108416332689586591351148, −3.95716768383396019529420793395, −2.75340677317787267014018672824, −1.91175357188616409849853761838, −0.982156667496023612952459407654,
0.982156667496023612952459407654, 1.91175357188616409849853761838, 2.75340677317787267014018672824, 3.95716768383396019529420793395, 4.72198108416332689586591351148, 5.33291685403171159343343644159, 6.10880545889739533497801635641, 7.04623862412338113697420853964, 7.65615425096118298080508648193, 8.354711609730548465701000111361
