| L(s) = 1 | + 17·7-s − 38·13-s − 107·19-s + 125·25-s + 289·31-s − 323·37-s + 142·43-s − 54·49-s − 182·61-s + 127·67-s + 271·73-s + 1.38e3·79-s − 646·91-s − 2.66e3·97-s + 1.80e3·103-s − 2.21e3·109-s + 1.33e3·121-s + 127-s + 131-s − 1.81e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.917·7-s − 0.810·13-s − 1.29·19-s + 25-s + 1.67·31-s − 1.43·37-s + 0.503·43-s − 0.157·49-s − 0.382·61-s + 0.231·67-s + 0.434·73-s + 1.97·79-s − 0.744·91-s − 2.78·97-s + 1.72·103-s − 1.94·109-s + 121-s + 0.000698·127-s + 0.000666·131-s − 1.18·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.200842725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200842725\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 17 T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )( 1 + 163 T + p^{3} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 110 T + p^{3} T^{2} )( 1 + 433 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 71 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 719 T + p^{3} T^{2} )( 1 + 901 T + p^{3} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 1007 T + p^{3} T^{2} )( 1 + 880 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )( 1 + 919 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 884 T + p^{3} T^{2} )( 1 - 503 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{3} T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \) |
| 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
−11.99142862389887593208134056956, −11.22188209352479935401790272884, −10.90418008308671326571106410172, −10.52499942711777554767940032264, −9.949373293350446087258603932355, −9.487188821549907607488293309340, −8.827910867186952998273668740303, −8.358715037796500356309811049923, −8.089563139334207279806188194282, −7.39244879268777096048117576275, −6.82138810938263262783923667325, −6.43273735845060132822148579398, −5.66614437925952831607898670657, −4.96848099143220736352785486481, −4.66329370800323904244372253302, −4.02240455283875516625804132796, −3.09199133555047477090152710617, −2.37165980107151361586766339429, −1.65308511388879026424102448556, −0.59445987539238523480168153030,
0.59445987539238523480168153030, 1.65308511388879026424102448556, 2.37165980107151361586766339429, 3.09199133555047477090152710617, 4.02240455283875516625804132796, 4.66329370800323904244372253302, 4.96848099143220736352785486481, 5.66614437925952831607898670657, 6.43273735845060132822148579398, 6.82138810938263262783923667325, 7.39244879268777096048117576275, 8.089563139334207279806188194282, 8.358715037796500356309811049923, 8.827910867186952998273668740303, 9.487188821549907607488293309340, 9.949373293350446087258603932355, 10.52499942711777554767940032264, 10.90418008308671326571106410172, 11.22188209352479935401790272884, 11.99142862389887593208134056956
