| L(s) = 1 | + 5-s + 3·9-s − 4·11-s + 4·13-s + 2·17-s + 4·19-s − 4·23-s − 4·29-s − 8·31-s − 6·37-s + 12·41-s − 16·43-s + 3·45-s + 4·47-s − 6·53-s − 4·55-s − 4·59-s − 2·61-s + 4·65-s − 8·67-s − 6·73-s + 32·83-s + 2·85-s − 6·89-s + 4·95-s + 28·97-s − 12·99-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 9-s − 1.20·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 0.742·29-s − 1.43·31-s − 0.986·37-s + 1.87·41-s − 2.43·43-s + 0.447·45-s + 0.583·47-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 0.256·61-s + 0.496·65-s − 0.977·67-s − 0.702·73-s + 3.51·83-s + 0.216·85-s − 0.635·89-s + 0.410·95-s + 2.84·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.329131834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.329131834\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.512509086409007994572809483649, −9.031498475158104799975340742924, −8.610886997194145311513936571838, −8.202510110926725375561992325429, −7.58908477500992717469597102699, −7.48000776587592498954195944030, −7.31993189195403858431227493458, −6.42879345535061633993076438240, −6.23736011171599227422803098777, −5.81719052051386365643483156906, −5.34286416306667222857382590396, −4.98457729243109005876415650039, −4.61450515674435707832336479544, −3.84035859024125448052399271146, −3.57719903384142266783470345619, −3.18034617942723019355588467044, −2.44272373468884620665589966239, −1.75196843018542689560959330384, −1.55673831530387375016807690684, −0.55182041369798724864729292085,
0.55182041369798724864729292085, 1.55673831530387375016807690684, 1.75196843018542689560959330384, 2.44272373468884620665589966239, 3.18034617942723019355588467044, 3.57719903384142266783470345619, 3.84035859024125448052399271146, 4.61450515674435707832336479544, 4.98457729243109005876415650039, 5.34286416306667222857382590396, 5.81719052051386365643483156906, 6.23736011171599227422803098777, 6.42879345535061633993076438240, 7.31993189195403858431227493458, 7.48000776587592498954195944030, 7.58908477500992717469597102699, 8.202510110926725375561992325429, 8.610886997194145311513936571838, 9.031498475158104799975340742924, 9.512509086409007994572809483649
