| L(s) = 1 | − 2-s + 2.68·3-s + 4-s + 1.11·5-s − 2.68·6-s − 8-s + 4.21·9-s − 1.11·10-s − 1.55·11-s + 2.68·12-s − 1.63·13-s + 2.98·15-s + 16-s + 17-s − 4.21·18-s + 4.18·19-s + 1.11·20-s + 1.55·22-s + 3.03·23-s − 2.68·24-s − 3.76·25-s + 1.63·26-s + 3.25·27-s + 9.69·29-s − 2.98·30-s + 0.426·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.55·3-s + 0.5·4-s + 0.497·5-s − 1.09·6-s − 0.353·8-s + 1.40·9-s − 0.351·10-s − 0.469·11-s + 0.775·12-s − 0.454·13-s + 0.771·15-s + 0.250·16-s + 0.242·17-s − 0.992·18-s + 0.959·19-s + 0.248·20-s + 0.331·22-s + 0.631·23-s − 0.548·24-s − 0.752·25-s + 0.321·26-s + 0.626·27-s + 1.80·29-s − 0.545·30-s + 0.0766·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.396929591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396929591\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 - 1.11T + 5T^{2} \) |
| 11 | \( 1 + 1.55T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 19 | \( 1 - 4.18T + 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 - 9.69T + 29T^{2} \) |
| 31 | \( 1 - 0.426T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 - 1.97T + 41T^{2} \) |
| 43 | \( 1 - 7.88T + 43T^{2} \) |
| 47 | \( 1 - 0.871T + 47T^{2} \) |
| 53 | \( 1 + 0.735T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 2.03T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 5.97T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 4.65T + 97T^{2} \) |
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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
−9.325253437545209506076028723767, −8.650269014502266726842498091551, −7.81919767824878132813062414677, −7.44400254672514127448834694990, −6.36505908558739419719394188291, −5.29956955048864321875160121323, −4.11574522997447862635833054821, −2.88841208630760641393529782561, −2.49465768259024927443632936864, −1.20554105093779654886793815090,
1.20554105093779654886793815090, 2.49465768259024927443632936864, 2.88841208630760641393529782561, 4.11574522997447862635833054821, 5.29956955048864321875160121323, 6.36505908558739419719394188291, 7.44400254672514127448834694990, 7.81919767824878132813062414677, 8.650269014502266726842498091551, 9.325253437545209506076028723767
