| L(s) = 1 | + (2.17 − 0.539i)5-s + i·7-s − 3.26·11-s + 0.340i·13-s + 5.75i·17-s − 6.49·19-s − 8.49i·23-s + (4.41 − 2.34i)25-s − 2·29-s + 8.34·31-s + (0.539 + 2.17i)35-s + 6.15i·37-s − 0.340·41-s + 8.68i·43-s − 49-s + ⋯ |
| L(s) = 1 | + (0.970 − 0.241i)5-s + 0.377i·7-s − 0.983·11-s + 0.0943i·13-s + 1.39i·17-s − 1.49·19-s − 1.77i·23-s + (0.883 − 0.468i)25-s − 0.371·29-s + 1.49·31-s + (0.0911 + 0.366i)35-s + 1.01i·37-s − 0.0531·41-s + 1.32i·43-s − 0.142·49-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
| \(L(1)\) |
\(\approx\) |
\(1.369141078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.369141078\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 - 0.340iT - 13T^{2} \) |
| 17 | \( 1 - 5.75iT - 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 + 8.49iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 - 6.15iT - 37T^{2} \) |
| 41 | \( 1 + 0.340T + 41T^{2} \) |
| 43 | \( 1 - 8.68iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 8.34iT - 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 1.50iT - 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 6.49iT - 97T^{2} \) |
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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
−8.367388382172336227814975765456, −8.070303834556386131000342000803, −6.69533728307119353523439405980, −6.29671196631810280582061242810, −5.65805715946511799742794886949, −4.73494713894398241687185551650, −4.20100606523034272543054441130, −2.75530945332803454447773198802, −2.33526006649043272180577202841, −1.24226186686596429636987061804,
0.34827993521245550401875743151, 1.75456698738196014767915361099, 2.52970940143904032667001866178, 3.33809198538436599046934102793, 4.41388773084599149520609839207, 5.25473706379676014533933749037, 5.72860488353987771124632414865, 6.67449363512174052982193238312, 7.23013185545057169214812397436, 7.972998533698043768588169166520
