| L(s) = 1 | + 2·4-s + 3·16-s + 8·17-s − 16·23-s + 4·25-s − 8·29-s − 16·43-s + 12·49-s + 8·53-s + 8·61-s + 12·64-s + 16·68-s − 32·92-s + 8·100-s + 8·101-s − 32·103-s − 24·113-s − 16·116-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4-s + 3/4·16-s + 1.94·17-s − 3.33·23-s + 4/5·25-s − 1.48·29-s − 2.43·43-s + 12/7·49-s + 1.09·53-s + 1.02·61-s + 3/2·64-s + 1.94·68-s − 3.33·92-s + 4/5·100-s + 0.796·101-s − 3.15·103-s − 2.25·113-s − 1.48·116-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.203720259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203720259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 1414 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13174 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 20566 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 12134 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 30166 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 11910 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.90043492964322665540792776585, −6.52959268461630362292830159002, −6.28627702241700423378615421842, −5.94202440577143058534471705652, −5.91176384280922585641463164091, −5.67705588196388609971948333683, −5.58682616470800108999038435411, −5.46861574243499322804283991877, −4.89390488221759809373207876682, −4.75739521901296692030258979398, −4.73763589401092088537985346974, −4.15611614972405100745365881388, −3.85492315404100634831648511588, −3.85439872593714081772190906812, −3.48422170140855559161992741632, −3.32265561939173602520037418595, −3.30004140918007563133705809734, −2.53104118696299928706669611890, −2.39408715863243546139022398651, −2.27157440887970660545576082852, −2.06125133608318246497319033480, −1.44680435929933217849711726220, −1.32686348417768137272425893545, −0.997291238901048976908257670895, −0.19196359566912129965311480479,
0.19196359566912129965311480479, 0.997291238901048976908257670895, 1.32686348417768137272425893545, 1.44680435929933217849711726220, 2.06125133608318246497319033480, 2.27157440887970660545576082852, 2.39408715863243546139022398651, 2.53104118696299928706669611890, 3.30004140918007563133705809734, 3.32265561939173602520037418595, 3.48422170140855559161992741632, 3.85439872593714081772190906812, 3.85492315404100634831648511588, 4.15611614972405100745365881388, 4.73763589401092088537985346974, 4.75739521901296692030258979398, 4.89390488221759809373207876682, 5.46861574243499322804283991877, 5.58682616470800108999038435411, 5.67705588196388609971948333683, 5.91176384280922585641463164091, 5.94202440577143058534471705652, 6.28627702241700423378615421842, 6.52959268461630362292830159002, 6.90043492964322665540792776585
