| L(s) = 1 | − 88·4-s + 3.76e3·16-s + 1.93e3·19-s + 1.44e4·31-s + 3.35e4·49-s + 8.54e4·61-s − 7.04e4·64-s − 1.70e5·76-s + 3.98e5·79-s − 6.77e3·109-s − 1.40e5·121-s − 1.27e6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.38e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2.75·4-s + 3.67·16-s + 1.23·19-s + 2.69·31-s + 1.99·49-s + 2.94·61-s − 2.14·64-s − 3.38·76-s + 7.18·79-s − 0.0546·109-s − 0.870·121-s − 7.41·124-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 + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 3.72·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.990002077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990002077\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + 11 p^{2} T^{2} + p^{10} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 2398 p T^{2} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 70102 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 692186 T^{2} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 57134 T^{2} + p^{10} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 484 T + p^{5} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 14654 p^{2} T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 10530298 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3608 T + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 83802314 T^{2} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 109744402 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 135962486 T^{2} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 367610894 T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 813621206 T^{2} + p^{10} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1399356598 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 21362 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 1504963814 T^{2} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2510746702 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 4138885586 T^{2} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 99616 T + p^{5} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 4615601606 T^{2} + p^{10} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 3347081102 T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 13052162114 T^{2} + p^{10} T^{4} )^{2} \) |
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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
−8.219374722619209288546357785252, −7.75031514100621964812471284677, −7.57033562413202487778818539905, −7.50573997781794297036214276731, −6.94029978667752341938988759531, −6.46657420796636436536923048516, −6.43925226554820944374815887077, −6.19803532883896065783702556961, −5.66610364390619884493196203967, −5.15234138160947870365311704329, −5.12969983867077137146956585999, −5.05836310530845657515505310733, −4.84137097099056610728782418347, −4.14703921449145171732802561158, −4.08612858644315460869876041254, −3.74542760636028952782164098529, −3.69212685895868846747300032892, −3.05737458051858383011178865773, −2.67137527212844741987893430929, −2.33386683535418631157816049537, −1.90837206004085068239217525252, −0.986285283335458837836901166433, −0.915415221838196968341604262363, −0.75443474112985258446962375667, −0.29539854319508773744557523772,
0.29539854319508773744557523772, 0.75443474112985258446962375667, 0.915415221838196968341604262363, 0.986285283335458837836901166433, 1.90837206004085068239217525252, 2.33386683535418631157816049537, 2.67137527212844741987893430929, 3.05737458051858383011178865773, 3.69212685895868846747300032892, 3.74542760636028952782164098529, 4.08612858644315460869876041254, 4.14703921449145171732802561158, 4.84137097099056610728782418347, 5.05836310530845657515505310733, 5.12969983867077137146956585999, 5.15234138160947870365311704329, 5.66610364390619884493196203967, 6.19803532883896065783702556961, 6.43925226554820944374815887077, 6.46657420796636436536923048516, 6.94029978667752341938988759531, 7.50573997781794297036214276731, 7.57033562413202487778818539905, 7.75031514100621964812471284677, 8.219374722619209288546357785252
