| L(s) = 1 | + 3-s + 2·5-s − 7-s − 9-s − 4·13-s + 2·15-s − 3·17-s + 19-s − 21-s − 2·23-s + 3·25-s − 7·29-s − 9·31-s − 2·35-s + 37-s − 4·39-s − 4·41-s + 4·43-s − 2·45-s + 18·47-s − 9·49-s − 3·51-s + 5·53-s + 57-s + 6·59-s − 17·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s − 1/3·9-s − 1.10·13-s + 0.516·15-s − 0.727·17-s + 0.229·19-s − 0.218·21-s − 0.417·23-s + 3/5·25-s − 1.29·29-s − 1.61·31-s − 0.338·35-s + 0.164·37-s − 0.640·39-s − 0.624·41-s + 0.609·43-s − 0.298·45-s + 2.62·47-s − 9/7·49-s − 0.420·51-s + 0.686·53-s + 0.132·57-s + 0.781·59-s − 2.17·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 34 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 156 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.34769984211322920105851890742, −7.15673988308875618716612719000, −6.99040546342831861828341047979, −6.55095159732329623875788620217, −6.02190306012742267495219298326, −5.81147874364961911527138518553, −5.52773904115290873515122314339, −5.11389584672322255327558410707, −4.91117125486459745062827588908, −4.19309987837688309586764055671, −3.99970415302905559884714436781, −3.70889574657385279929283056347, −2.96107292025292260664613986877, −2.86166812998242893449730268559, −2.32748753202648740434026122129, −2.15524747745675525785596233658, −1.57384052323260492824510279601, −1.11865406624299135599817496595, 0, 0,
1.11865406624299135599817496595, 1.57384052323260492824510279601, 2.15524747745675525785596233658, 2.32748753202648740434026122129, 2.86166812998242893449730268559, 2.96107292025292260664613986877, 3.70889574657385279929283056347, 3.99970415302905559884714436781, 4.19309987837688309586764055671, 4.91117125486459745062827588908, 5.11389584672322255327558410707, 5.52773904115290873515122314339, 5.81147874364961911527138518553, 6.02190306012742267495219298326, 6.55095159732329623875788620217, 6.99040546342831861828341047979, 7.15673988308875618716612719000, 7.34769984211322920105851890742
