| L(s) = 1 | + 6·2-s − 9·3-s + 32·4-s − 6·5-s − 54·6-s + 360·8-s − 36·10-s + 564·11-s − 288·12-s + 1.27e3·13-s + 54·15-s + 2.16e3·16-s − 882·17-s + 556·19-s − 192·20-s + 3.38e3·22-s + 840·23-s − 3.24e3·24-s + 3.12e3·25-s + 7.65e3·26-s + 729·27-s + 9.27e3·29-s + 324·30-s − 4.40e3·31-s + 1.15e4·32-s − 5.07e3·33-s − 5.29e3·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 4-s − 0.107·5-s − 0.612·6-s + 1.98·8-s − 0.113·10-s + 1.40·11-s − 0.577·12-s + 2.09·13-s + 0.0619·15-s + 2.10·16-s − 0.740·17-s + 0.353·19-s − 0.107·20-s + 1.49·22-s + 0.331·23-s − 1.14·24-s + 25-s + 2.22·26-s + 0.192·27-s + 2.04·29-s + 0.0657·30-s − 0.822·31-s + 1.98·32-s − 0.811·33-s − 0.785·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.836026128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.836026128\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T + p^{2} T^{2} - 3 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T - 3089 T^{2} + 6 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 564 T + 157045 T^{2} - 564 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 638 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 882 T - 641933 T^{2} + 882 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 556 T - 2166963 T^{2} - 556 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 840 T - 5730743 T^{2} - 840 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4638 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 4400 T - 9269151 T^{2} + 4400 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2410 T - 63535857 T^{2} - 2410 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6870 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9644 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18672 T + 119298577 T^{2} - 18672 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 33750 T + 720867007 T^{2} + 33750 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18084 T - 387893243 T^{2} - 18084 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 39758 T + 736102263 T^{2} + 39758 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 23068 T - 817992483 T^{2} - 23068 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4248 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 41110 T - 383039493 T^{2} - 41110 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 21920 T - 2596569999 T^{2} + 21920 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 82452 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 94086 T + 3268115947 T^{2} - 94086 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 49442 T + p^{5} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.45372071946020634920031330111, −11.90778884714561719284705689024, −11.29464208594509385140633168421, −11.12968398966023896321589912350, −10.54782916278907889013904056008, −10.27498540426477541396479886761, −9.022008078519414652505211453499, −8.960664635691604812565763139021, −8.071052544266762428696511892278, −7.49777515162677680518887614385, −6.63027983461981637133109721718, −6.48155236667716137173350538658, −5.97821623296091101697321192634, −4.98185980028052284672400834756, −4.68743937478067300448234885841, −3.81682250413544175813984633584, −3.53957087168519591641235408804, −2.38050390290165697923108390506, −1.14587030938156797064657435408, −1.13870219126878295669603293487,
1.13870219126878295669603293487, 1.14587030938156797064657435408, 2.38050390290165697923108390506, 3.53957087168519591641235408804, 3.81682250413544175813984633584, 4.68743937478067300448234885841, 4.98185980028052284672400834756, 5.97821623296091101697321192634, 6.48155236667716137173350538658, 6.63027983461981637133109721718, 7.49777515162677680518887614385, 8.071052544266762428696511892278, 8.960664635691604812565763139021, 9.022008078519414652505211453499, 10.27498540426477541396479886761, 10.54782916278907889013904056008, 11.12968398966023896321589912350, 11.29464208594509385140633168421, 11.90778884714561719284705689024, 12.45372071946020634920031330111
