| L(s) = 1 | + i·3-s + 2.23·5-s + i·7-s − 9-s + 2·11-s − 4.47i·13-s + 2.23i·15-s − 6.47i·17-s + 2·19-s − 21-s − 4i·23-s + 5.00·25-s − i·27-s + 8.47·29-s + 0.472·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.999·5-s + 0.377i·7-s − 0.333·9-s + 0.603·11-s − 1.24i·13-s + 0.577i·15-s − 1.56i·17-s + 0.458·19-s − 0.218·21-s − 0.834i·23-s + 1.00·25-s − 0.192i·27-s + 1.57·29-s + 0.0847·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166450287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166450287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 0.472T + 31T^{2} \) |
| 37 | \( 1 + 2.47iT - 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 - 2.47iT - 43T^{2} \) |
| 47 | \( 1 - 6.47iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 1.52iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.52iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 4.94iT - 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 + 3.52iT - 97T^{2} \) |
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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
−9.365662707639186489561039894012, −8.813741533408734104095286023879, −7.87381300535201958282009287904, −6.79758228274833664663064268498, −6.02589171913680573336794383658, −5.21822241519116633952655098494, −4.59066596833036635818773195792, −3.14043716612225681638802522076, −2.55117428671047054305860631940, −0.950268170574869678856090109351,
1.31107122907419097395558461800, 1.98393875322555708005438014281, 3.32299895968630026163813232651, 4.36479531582599764469203548637, 5.40008369206042728836092980195, 6.44648988037086414665311803493, 6.62754550826837263948662540680, 7.73756605431910636080781919973, 8.667292505982898318200228799091, 9.264208481133307120037657457469
