| L(s) = 1 | + 2i·7-s − 5·11-s + 5i·17-s − 5·19-s − 6i·23-s + 4·29-s + 10·31-s + 10i·37-s − 5·41-s + 4i·43-s − 8i·47-s + 3·49-s + 10i·53-s − 10·61-s − 3i·67-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s − 1.50·11-s + 1.21i·17-s − 1.14·19-s − 1.25i·23-s + 0.742·29-s + 1.79·31-s + 1.64i·37-s − 0.780·41-s + 0.609i·43-s − 1.16i·47-s + 0.428·49-s + 1.37i·53-s − 1.28·61-s − 0.366i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3588927583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3588927583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 5iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.001200429110217781127095934640, −6.86321698490405171872989459535, −6.25539741310602627489639645224, −5.69716245654546124102327817574, −4.73195858614391250243416121206, −4.35102195707219607805513246131, −2.97590832759699342276580701877, −2.60039703955394184412487515662, −1.58237202439360885511901440437, −0.096609520968591597397577249996,
0.946771510326957582099687517257, 2.26004773416192368350231326520, 2.89849467608506502650243031881, 3.85406954554083744119136396546, 4.67439936844427499058545876276, 5.22633083961688524811836886884, 6.05001814601367796966109426352, 6.89227647999716666087434851707, 7.48120076200818950937611115087, 8.025598430871014784734312261274
