| L(s) = 1 | + 5-s − 3·11-s − 6·13-s − 5·17-s − 19-s − 7·23-s − 4·25-s + 2·29-s − 5·31-s − 3·37-s − 2·41-s − 4·43-s − 5·47-s − 53-s − 3·55-s + 15·59-s − 5·61-s − 6·65-s − 9·67-s − 7·73-s − 79-s + 12·83-s − 5·85-s + 7·89-s − 95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.904·11-s − 1.66·13-s − 1.21·17-s − 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.371·29-s − 0.898·31-s − 0.493·37-s − 0.312·41-s − 0.609·43-s − 0.729·47-s − 0.137·53-s − 0.404·55-s + 1.95·59-s − 0.640·61-s − 0.744·65-s − 1.09·67-s − 0.819·73-s − 0.112·79-s + 1.31·83-s − 0.542·85-s + 0.741·89-s − 0.102·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3682960629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3682960629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.16584605855201, −14.72346437298181, −14.14893252028528, −13.55158450963127, −13.15541046441091, −12.60636904674830, −11.94352331569466, −11.62859221073525, −10.75830811630059, −10.28621510432262, −9.869060102343081, −9.337613937258824, −8.642308322331607, −8.023558467077805, −7.513478539440773, −6.899434734675473, −6.292412750617588, −5.588357157682364, −5.054618727776293, −4.485637998554770, −3.766972621333801, −2.829773799802868, −2.185047850671946, −1.806196979329178, −0.2137652860221413,
0.2137652860221413, 1.806196979329178, 2.185047850671946, 2.829773799802868, 3.766972621333801, 4.485637998554770, 5.054618727776293, 5.588357157682364, 6.292412750617588, 6.899434734675473, 7.513478539440773, 8.023558467077805, 8.642308322331607, 9.337613937258824, 9.869060102343081, 10.28621510432262, 10.75830811630059, 11.62859221073525, 11.94352331569466, 12.60636904674830, 13.15541046441091, 13.55158450963127, 14.14893252028528, 14.72346437298181, 15.16584605855201
