L(s) = 1 | + (2.17 − 0.539i)5-s − i·7-s + 5.41·11-s − 4.34i·13-s − 1.07i·17-s + 4.34·19-s − 6.34i·23-s + (4.41 − 2.34i)25-s − 8.83·29-s + 4.34·31-s + (−0.539 − 2.17i)35-s − 8.68i·37-s − 8.34·41-s + 6.15i·43-s − 6.83i·47-s + ⋯ |
L(s) = 1 | + (0.970 − 0.241i)5-s − 0.377i·7-s + 1.63·11-s − 1.20i·13-s − 0.261i·17-s + 0.995·19-s − 1.32i·23-s + (0.883 − 0.468i)25-s − 1.64·29-s + 0.779·31-s + (−0.0911 − 0.366i)35-s − 1.42i·37-s − 1.30·41-s + 0.938i·43-s − 0.997i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.638906129\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.638906129\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.17 + 0.539i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 4.34iT - 13T^{2} \) |
| 17 | \( 1 + 1.07iT - 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 6.34iT - 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 + 8.68iT - 37T^{2} \) |
| 41 | \( 1 + 8.34T + 41T^{2} \) |
| 43 | \( 1 - 6.15iT - 43T^{2} \) |
| 47 | \( 1 + 6.83iT - 47T^{2} \) |
| 53 | \( 1 - 6.18iT - 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 - 0.680T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−8.148599214790657913645503567989, −7.24507833932134258284056302769, −6.64360979679106640009537990562, −5.86720930232803946798648471332, −5.29937619009144064015935405938, −4.37360887843136529811928138406, −3.55639171219658295957004474112, −2.64387412583295144009033824384, −1.55799116004474242025304180930, −0.72174761720976282092262123330,
1.45849441114014214021733887501, 1.80008523886097250863541947771, 3.11292514693152223794055164383, 3.80530723647412184223168735575, 4.78278414054773851838106802743, 5.59805946261379186369097042657, 6.27119531021125502339769991617, 6.82758198713030061528495421114, 7.50074496944604088833127390291, 8.637389190950124826468984724803