| L(s) = 1 | + 416·7-s + 422·9-s − 2.55e3·17-s − 6.43e3·23-s + 6.05e3·25-s − 5.24e3·31-s − 340·41-s + 64·47-s + 9.61e4·49-s + 1.75e5·63-s − 5.71e4·71-s + 1.07e5·73-s − 1.38e5·79-s + 1.19e5·81-s + 2.53e5·89-s + 1.24e5·97-s + 2.16e5·103-s + 3.15e4·113-s − 1.06e6·119-s + 3.48e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.07e6·153-s + ⋯ |
| L(s) = 1 | + 3.20·7-s + 1.73·9-s − 2.14·17-s − 2.53·23-s + 1.93·25-s − 0.980·31-s − 0.0315·41-s + 0.00422·47-s + 5.72·49-s + 5.57·63-s − 1.34·71-s + 2.35·73-s − 2.49·79-s + 2.01·81-s + 3.39·89-s + 1.34·97-s + 2.01·103-s + 0.232·113-s − 6.88·119-s + 0.216·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 3.72·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.390918505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.390918505\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 422 T^{2} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 6054 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 208 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 34806 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 260950 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 1278 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 3715654 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3216 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 32507574 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2624 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49234150 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 170 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 103108298 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 32 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 344527302 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 290741802 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 1450119158 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2269936678 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 28592 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 53670 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 69152 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6449241286 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 126806 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 62290 T + p^{5} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.44780104122107624251360267314, −10.88495356846117377540599731848, −10.41942688099827209827660002641, −10.37388543537296633672628988916, −9.295951346106431403643187221709, −8.964314878010123573620486359725, −8.213689189214615869069406915667, −8.199996097255667135006589809301, −7.39012552741065834452616996736, −7.20717554339054755910560444093, −6.47018700238342486898090336850, −5.76023526312913551771022503101, −4.79780580665124183272760649110, −4.77314788858617521228719445927, −4.34331928085504995350318672462, −3.73646627231211802975509518765, −2.19101098040255214106304424519, −1.97285245975287933786649133801, −1.49708318724447404581751342634, −0.66928785120847078509002546052,
0.66928785120847078509002546052, 1.49708318724447404581751342634, 1.97285245975287933786649133801, 2.19101098040255214106304424519, 3.73646627231211802975509518765, 4.34331928085504995350318672462, 4.77314788858617521228719445927, 4.79780580665124183272760649110, 5.76023526312913551771022503101, 6.47018700238342486898090336850, 7.20717554339054755910560444093, 7.39012552741065834452616996736, 8.199996097255667135006589809301, 8.213689189214615869069406915667, 8.964314878010123573620486359725, 9.295951346106431403643187221709, 10.37388543537296633672628988916, 10.41942688099827209827660002641, 10.88495356846117377540599731848, 11.44780104122107624251360267314
