| L(s) = 1 | + 2.01·2-s + 1.86·3-s + 2.07·4-s − 0.244·5-s + 3.75·6-s − 7-s + 0.158·8-s + 0.462·9-s − 0.493·10-s + 2.58·11-s + 3.86·12-s − 6.33·13-s − 2.01·14-s − 0.454·15-s − 3.83·16-s − 0.782·17-s + 0.933·18-s + 5.67·19-s − 0.508·20-s − 1.86·21-s + 5.21·22-s + 5.86·23-s + 0.295·24-s − 4.94·25-s − 12.8·26-s − 4.72·27-s − 2.07·28-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.07·3-s + 1.03·4-s − 0.109·5-s + 1.53·6-s − 0.377·7-s + 0.0561·8-s + 0.154·9-s − 0.156·10-s + 0.778·11-s + 1.11·12-s − 1.75·13-s − 0.539·14-s − 0.117·15-s − 0.959·16-s − 0.189·17-s + 0.220·18-s + 1.30·19-s − 0.113·20-s − 0.406·21-s + 1.11·22-s + 1.22·23-s + 0.0603·24-s − 0.988·25-s − 2.51·26-s − 0.908·27-s − 0.392·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.092578689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.092578689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 2.01T + 2T^{2} \) |
| 3 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 + 0.244T + 5T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 + 6.33T + 13T^{2} \) |
| 17 | \( 1 + 0.782T + 17T^{2} \) |
| 19 | \( 1 - 5.67T + 19T^{2} \) |
| 23 | \( 1 - 5.86T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 0.680T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 8.88T + 59T^{2} \) |
| 61 | \( 1 + 9.26T + 61T^{2} \) |
| 67 | \( 1 - 9.47T + 67T^{2} \) |
| 71 | \( 1 - 6.21T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 4.05T + 83T^{2} \) |
| 89 | \( 1 + 6.12T + 89T^{2} \) |
| 97 | \( 1 - 6.54T + 97T^{2} \) |
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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
−12.13448744382071344717011467585, −11.31306343887790518395001543688, −9.647489797353992897380013197700, −9.161629080693397784256888495126, −7.73753409763952379454113442994, −6.83221139194141177972383629950, −5.54542361702429454275172415652, −4.46345831168935850252696587765, −3.33620011226421082636492671267, −2.51833126683717104335404106352,
2.51833126683717104335404106352, 3.33620011226421082636492671267, 4.46345831168935850252696587765, 5.54542361702429454275172415652, 6.83221139194141177972383629950, 7.73753409763952379454113442994, 9.161629080693397784256888495126, 9.647489797353992897380013197700, 11.31306343887790518395001543688, 12.13448744382071344717011467585
