| L(s) = 1 | + 2-s − 4-s + (−1.78 + 1.35i)5-s + 3.46i·7-s − 3·8-s + (−1.78 + 1.35i)10-s − 6.16i·11-s + 2.70i·13-s + 3.46i·14-s − 16-s − 2·17-s − 3.46i·19-s + (1.78 − 1.35i)20-s − 6.16i·22-s − 3.46i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.5·4-s + (−0.796 + 0.604i)5-s + 1.30i·7-s − 1.06·8-s + (−0.563 + 0.427i)10-s − 1.85i·11-s + 0.750i·13-s + 0.925i·14-s − 0.250·16-s − 0.485·17-s − 0.794i·19-s + (0.398 − 0.302i)20-s − 1.31i·22-s − 0.722i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6378345729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6378345729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.78 - 1.35i)T \) |
| 29 | \( 1 + (-4.12 + 3.46i)T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 6.16iT - 11T^{2} \) |
| 13 | \( 1 - 2.70iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 31 | \( 1 + 0.759iT - 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 5.40iT - 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 - 0.684T + 47T^{2} \) |
| 53 | \( 1 - 2.70iT - 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 12.3iT - 61T^{2} \) |
| 67 | \( 1 - 1.94iT - 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 - 7.68iT - 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 5.40iT - 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−8.985591462974796574518378192215, −8.797899797175791452450019134285, −7.984989928991688234917668478499, −6.55530885698656436411880909652, −6.11339804710674553480174302421, −5.11838058514613447582508660963, −4.23765173728633795794916638737, −3.25085509507549939780789668750, −2.58103571066133670247223067530, −0.22737874140136406708681448860,
1.33330068324424008595052916678, 3.18960377182551593238955110876, 4.11123116851441865629012377638, 4.58259226043927656285763485837, 5.32241656275636039305172397398, 6.65246216108314240589369818666, 7.47602481311772699374186014757, 8.095397153739050492988783050261, 9.096233155695661601024797592859, 9.955421248272266139841128225692
