| 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
−9.124794221872116073003463240844, −8.789616090104045527964856577049, −7.76612767480301943300680066515, −6.73999239465243409347204606476, −5.92104777030879332334984403212, −4.64622862781921420867562534243, −3.62735962932438679534243527039, −2.76813292916274978802455244177, −2.03829401380890338725458088501, −0.35420853440194243953923057116,
1.16923579954615080831175118171, 2.41494561031306130401015048528, 4.01603530283211187039009158866, 4.55362670802625689011220061591, 5.45676042879623974439533014159, 6.85062451753392867027536244855, 7.16165388409966698832172935135, 8.458817569423969188568435150628, 8.706375114149408590148870097491, 9.670128744536759041032034745284
