| L(s) = 1 | + 2-s − 1.68·3-s + 4-s + 5-s − 1.68·6-s − 7-s + 8-s − 0.154·9-s + 10-s − 1.68·12-s − 0.0425·13-s − 14-s − 1.68·15-s + 16-s + 4.72·17-s − 0.154·18-s + 1.71·19-s + 20-s + 1.68·21-s − 6.95·23-s − 1.68·24-s + 25-s − 0.0425·26-s + 5.32·27-s − 28-s + 1.34·29-s − 1.68·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.973·3-s + 0.5·4-s + 0.447·5-s − 0.688·6-s − 0.377·7-s + 0.353·8-s − 0.0514·9-s + 0.316·10-s − 0.486·12-s − 0.0118·13-s − 0.267·14-s − 0.435·15-s + 0.250·16-s + 1.14·17-s − 0.0363·18-s + 0.393·19-s + 0.223·20-s + 0.368·21-s − 1.45·23-s − 0.344·24-s + 0.200·25-s − 0.00834·26-s + 1.02·27-s − 0.188·28-s + 0.249·29-s − 0.307·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.210406998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210406998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.68T + 3T^{2} \) |
| 13 | \( 1 + 0.0425T + 13T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 - 1.71T + 19T^{2} \) |
| 23 | \( 1 + 6.95T + 23T^{2} \) |
| 29 | \( 1 - 1.34T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 - 4.59T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 6.39T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.98T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 2.93T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.78471531827781674216789124522, −6.70602702613022865935704283486, −6.19014255594198576311851881999, −5.83575129502058173080393898736, −5.05346253870865437533479274952, −4.50976457066452219525786613816, −3.45217129864692571215120697000, −2.84553053769287833215163476179, −1.76474703321371671886084958456, −0.68861058607790811979049841117,
0.68861058607790811979049841117, 1.76474703321371671886084958456, 2.84553053769287833215163476179, 3.45217129864692571215120697000, 4.50976457066452219525786613816, 5.05346253870865437533479274952, 5.83575129502058173080393898736, 6.19014255594198576311851881999, 6.70602702613022865935704283486, 7.78471531827781674216789124522
