| L(s) = 1 | − 6·3-s + 27·9-s − 20·11-s + 46·25-s − 108·27-s + 120·33-s − 2·49-s + 20·59-s − 100·73-s − 276·75-s + 405·81-s + 268·83-s − 380·97-s − 540·99-s − 172·107-s + 58·121-s + 127-s + 131-s + 137-s + 139-s + 12·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + ⋯ |
| L(s) = 1 | − 2·3-s + 3·9-s − 1.81·11-s + 1.83·25-s − 4·27-s + 3.63·33-s − 0.0408·49-s + 0.338·59-s − 1.36·73-s − 3.67·75-s + 5·81-s + 3.22·83-s − 3.91·97-s − 5.45·99-s − 1.60·107-s + 0.479·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 4/49·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6031511381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6031511381\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 818 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 478 T^{2} + p^{4} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3218 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 9118 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 134 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 190 T + p^{2} T^{2} )^{2} \) |
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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
−11.09174178305860298462343050747, −10.93368520438387004591021462384, −10.53614641723231014480380593918, −10.34913422091247427965347447688, −9.576548967975523669263158284831, −9.399297274426748358735490416839, −8.379158771935378055289746023366, −8.114697918700641662948794273585, −7.26712211626221721558232007861, −7.17724177288283887180247738741, −6.50970445320169563016956910771, −6.04178419826801823165719356188, −5.36886853699501501833767542623, −5.20177340411214930889547021371, −4.66827396284034161390610368911, −4.10420312669891528877954122971, −3.15621449632522271563789713871, −2.34804608197085442419145609387, −1.30785955757393758233558516199, −0.42412456809419320682839736612,
0.42412456809419320682839736612, 1.30785955757393758233558516199, 2.34804608197085442419145609387, 3.15621449632522271563789713871, 4.10420312669891528877954122971, 4.66827396284034161390610368911, 5.20177340411214930889547021371, 5.36886853699501501833767542623, 6.04178419826801823165719356188, 6.50970445320169563016956910771, 7.17724177288283887180247738741, 7.26712211626221721558232007861, 8.114697918700641662948794273585, 8.379158771935378055289746023366, 9.399297274426748358735490416839, 9.576548967975523669263158284831, 10.34913422091247427965347447688, 10.53614641723231014480380593918, 10.93368520438387004591021462384, 11.09174178305860298462343050747
