| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 2·7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s + 13-s + 2·14-s − 15-s + 16-s − 2·17-s + 18-s − 6·19-s + 20-s − 2·21-s − 4·22-s + 7·23-s − 24-s + 25-s + 26-s − 27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.436·21-s − 0.852·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.73776394118223, −12.68737274271089, −11.86762708640271, −11.45534740940542, −10.98681048545891, −10.73232188544957, −10.30655347728149, −9.872899448409542, −9.110986616469560, −8.509862293991117, −8.397191328546574, −7.611095739824629, −7.137123165891999, −6.686792123655263, −6.305321143377508, −5.589708788428013, −5.263009767617685, −4.920698914129289, −4.461754917190583, −3.811546702724079, −3.270031224839612, −2.484050593156250, −2.164311824142015, −1.566003588938769, −0.8226089253914869, 0,
0.8226089253914869, 1.566003588938769, 2.164311824142015, 2.484050593156250, 3.270031224839612, 3.811546702724079, 4.461754917190583, 4.920698914129289, 5.263009767617685, 5.589708788428013, 6.305321143377508, 6.686792123655263, 7.137123165891999, 7.611095739824629, 8.397191328546574, 8.509862293991117, 9.110986616469560, 9.872899448409542, 10.30655347728149, 10.73232188544957, 10.98681048545891, 11.45534740940542, 11.86762708640271, 12.68737274271089, 12.73776394118223
