| L(s) = 1 | + 3-s + 3·5-s − 5·7-s + 9·11-s + 3·15-s + 6·17-s − 2·19-s − 5·21-s − 12·23-s + 25-s − 27-s + 18·29-s + 31-s + 9·33-s − 15·35-s − 2·37-s + 18·49-s + 6·51-s + 9·53-s + 27·55-s − 2·57-s + 3·59-s + 12·61-s − 12·69-s + 12·73-s + 75-s − 45·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 1.88·7-s + 2.71·11-s + 0.774·15-s + 1.45·17-s − 0.458·19-s − 1.09·21-s − 2.50·23-s + 1/5·25-s − 0.192·27-s + 3.34·29-s + 0.179·31-s + 1.56·33-s − 2.53·35-s − 0.328·37-s + 18/7·49-s + 0.840·51-s + 1.23·53-s + 3.64·55-s − 0.264·57-s + 0.390·59-s + 1.53·61-s − 1.44·69-s + 1.40·73-s + 0.115·75-s − 5.12·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.670896816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.670896816\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
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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
−9.901703733266895180542999915957, −9.535065975846172405240958564469, −9.087756639684113404695772944098, −8.734897173170311866222437243811, −8.173762841779971970736320018325, −8.144975251171845251372689290064, −6.98180667752108816020964141101, −6.87261802571745326498665855103, −6.59305759877207047667008374207, −6.09462272215053025909891215494, −5.73708288278090374630242645945, −5.64802043621953387760691101209, −4.43509131294369892277703701017, −4.20264034259411299963558811676, −3.62173566644293999277133840253, −3.35279970094034003351239438763, −2.63443780697404757407179083800, −2.20306919143193019036465934251, −1.44498877374486520915247016129, −0.819391462600540293555126408636,
0.819391462600540293555126408636, 1.44498877374486520915247016129, 2.20306919143193019036465934251, 2.63443780697404757407179083800, 3.35279970094034003351239438763, 3.62173566644293999277133840253, 4.20264034259411299963558811676, 4.43509131294369892277703701017, 5.64802043621953387760691101209, 5.73708288278090374630242645945, 6.09462272215053025909891215494, 6.59305759877207047667008374207, 6.87261802571745326498665855103, 6.98180667752108816020964141101, 8.144975251171845251372689290064, 8.173762841779971970736320018325, 8.734897173170311866222437243811, 9.087756639684113404695772944098, 9.535065975846172405240958564469, 9.901703733266895180542999915957
