L(s) = 1 | + 2-s − 3.30·3-s + 4-s + 0.376·5-s − 3.30·6-s + 7-s + 8-s + 7.92·9-s + 0.376·10-s + 5.68·11-s − 3.30·12-s + 14-s − 1.24·15-s + 16-s + 1.24·17-s + 7.92·18-s + 1.62·19-s + 0.376·20-s − 3.30·21-s + 5.68·22-s + 4.92·23-s − 3.30·24-s − 4.85·25-s − 16.2·27-s + 28-s − 4.61·29-s − 1.24·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.90·3-s + 0.5·4-s + 0.168·5-s − 1.34·6-s + 0.377·7-s + 0.353·8-s + 2.64·9-s + 0.119·10-s + 1.71·11-s − 0.954·12-s + 0.267·14-s − 0.321·15-s + 0.250·16-s + 0.302·17-s + 1.86·18-s + 0.372·19-s + 0.0842·20-s − 0.721·21-s + 1.21·22-s + 1.02·23-s − 0.674·24-s − 0.971·25-s − 3.13·27-s + 0.188·28-s − 0.856·29-s − 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.893716372\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.893716372\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 - 0.376T + 5T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 4.92T + 23T^{2} \) |
| 29 | \( 1 + 4.61T + 29T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 + 3.24T + 43T^{2} \) |
| 47 | \( 1 - 2.31T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.30T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 0.317T + 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 - 9.74T + 83T^{2} \) |
| 89 | \( 1 + 9.24T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.330228903821700802838571582625, −7.88336525186266056407670575718, −6.92376247078996432905454173649, −6.59071624267587586238009960087, −5.63456710653650521273443694650, −5.28254029208122141489039151267, −4.26814079445525864522775297227, −3.69953076808292878029896517131, −1.83698409748401742657220602360, −0.948032633732754021203303713985,
0.948032633732754021203303713985, 1.83698409748401742657220602360, 3.69953076808292878029896517131, 4.26814079445525864522775297227, 5.28254029208122141489039151267, 5.63456710653650521273443694650, 6.59071624267587586238009960087, 6.92376247078996432905454173649, 7.88336525186266056407670575718, 9.330228903821700802838571582625