| L(s) = 1 | + 2i·2-s + 3·3-s − 4·4-s − 10i·5-s + 6i·6-s − 8i·7-s − 8i·8-s + 9·9-s + 20·10-s + 40i·11-s − 12·12-s + 16·14-s − 30i·15-s + 16·16-s − 130·17-s + 18i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.894i·5-s + 0.408i·6-s − 0.431i·7-s − 0.353i·8-s + 0.333·9-s + 0.632·10-s + 1.09i·11-s − 0.288·12-s + 0.305·14-s − 0.516i·15-s + 0.250·16-s − 1.85·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.482392771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482392771\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 10iT - 125T^{2} \) |
| 7 | \( 1 + 8iT - 343T^{2} \) |
| 11 | \( 1 - 40iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 130T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 18T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 74iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 362iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 76T + 7.95e4T^{2} \) |
| 47 | \( 1 + 452iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 382T + 1.48e5T^{2} \) |
| 59 | \( 1 - 464iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 358T + 2.26e5T^{2} \) |
| 67 | \( 1 - 700iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 748iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.05e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 976T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 386iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 614iT - 9.12e5T^{2} \) |
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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
−9.670128744536759041032034745284, −8.706375114149408590148870097491, −8.458817569423969188568435150628, −7.16165388409966698832172935135, −6.85062451753392867027536244855, −5.45676042879623974439533014159, −4.55362670802625689011220061591, −4.01603530283211187039009158866, −2.41494561031306130401015048528, −1.16923579954615080831175118171,
0.35420853440194243953923057116, 2.03829401380890338725458088501, 2.76813292916274978802455244177, 3.62735962932438679534243527039, 4.64622862781921420867562534243, 5.92104777030879332334984403212, 6.73999239465243409347204606476, 7.76612767480301943300680066515, 8.789616090104045527964856577049, 9.124794221872116073003463240844
