| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.82i·8-s − 3.78i·11-s + 4.00·16-s − 8.02i·17-s − 8.34·19-s − 5.34·22-s − 5·25-s − 5.65i·32-s − 11.3·34-s + 11.8i·38-s + 0.460i·41-s − 2.34·43-s + 7.56i·44-s + ⋯ |
| L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 1.00i·8-s − 1.14i·11-s + 1.00·16-s − 1.94i·17-s − 1.91·19-s − 1.14·22-s − 25-s − 1.00i·32-s − 1.94·34-s + 1.91i·38-s + 0.0719i·41-s − 0.358·43-s + 1.14i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.824870i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.824870i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3.78iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 8.02iT - 17T^{2} \) |
| 19 | \( 1 + 8.34T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 0.460iT - 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 12.2iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 - 19.6T + 97T^{2} \) |
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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
−10.25925732113206751279818468607, −9.320229397252411977138110405342, −8.642838249226461144087925557949, −7.74291470598763338407748441240, −6.40152793338085444395816460526, −5.34432275054766543978739586691, −4.33527821899086654859756035047, −3.25221440944269677235185566508, −2.18237577280303330503605838984, −0.43824990547743013496252297095,
1.92210431985025915873400622476, 3.89298545530081624117668937954, 4.51615354657652496836115593356, 5.82310435506547778896802177956, 6.47900419701703289200527048364, 7.46643886160990206359066577902, 8.301192212782763618557322012454, 9.010201116591926500470581077527, 10.11590133203919094166445563662, 10.61341225303751529165671074484
