| L(s) = 1 | − 2·3-s + 2·5-s + 6·7-s + 2·9-s + 6·13-s − 4·15-s + 2·17-s − 8·19-s − 12·21-s + 2·23-s − 25-s − 6·27-s + 12·35-s − 2·37-s − 12·39-s − 20·41-s − 10·43-s + 4·45-s + 6·47-s + 18·49-s − 4·51-s − 10·53-s + 16·57-s + 24·59-s + 4·61-s + 12·63-s + 12·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 2.26·7-s + 2/3·9-s + 1.66·13-s − 1.03·15-s + 0.485·17-s − 1.83·19-s − 2.61·21-s + 0.417·23-s − 1/5·25-s − 1.15·27-s + 2.02·35-s − 0.328·37-s − 1.92·39-s − 3.12·41-s − 1.52·43-s + 0.596·45-s + 0.875·47-s + 18/7·49-s − 0.560·51-s − 1.37·53-s + 2.11·57-s + 3.12·59-s + 0.512·61-s + 1.51·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243471294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243471294\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−12.95925770827375997058743013906, −12.84139341637347290761539981621, −11.69293023331710011895744125099, −11.63382711835295488583924754104, −11.28655009197773379600839819346, −10.74819187576907318581233985160, −10.22712027625269440624422582566, −9.945546337542022813772769662669, −8.772467860817053579338485519718, −8.488576150638774646669330552913, −8.206353204226717108679830505973, −7.28484508312509426651299023159, −6.58323582471142846291919349073, −6.12815307755934488180846606296, −5.28864812120296300610821443940, −5.28133571641083190207100039484, −4.38004981953623604220263144997, −3.65475271465668673728337717689, −1.95726066495511050852451073507, −1.51748122480919915396563750512,
1.51748122480919915396563750512, 1.95726066495511050852451073507, 3.65475271465668673728337717689, 4.38004981953623604220263144997, 5.28133571641083190207100039484, 5.28864812120296300610821443940, 6.12815307755934488180846606296, 6.58323582471142846291919349073, 7.28484508312509426651299023159, 8.206353204226717108679830505973, 8.488576150638774646669330552913, 8.772467860817053579338485519718, 9.945546337542022813772769662669, 10.22712027625269440624422582566, 10.74819187576907318581233985160, 11.28655009197773379600839819346, 11.63382711835295488583924754104, 11.69293023331710011895744125099, 12.84139341637347290761539981621, 12.95925770827375997058743013906
