| L(s) = 1 | + 3-s + 4-s + 2·5-s + 7-s + 9-s + 12-s + 2·15-s + 16-s − 12·17-s + 2·20-s + 21-s + 3·25-s + 27-s + 28-s + 2·35-s + 36-s − 20·37-s − 12·41-s − 8·43-s + 2·45-s + 48-s + 49-s − 12·51-s − 24·59-s + 2·60-s + 63-s + 64-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.288·12-s + 0.516·15-s + 1/4·16-s − 2.91·17-s + 0.447·20-s + 0.218·21-s + 3/5·25-s + 0.192·27-s + 0.188·28-s + 0.338·35-s + 1/6·36-s − 3.28·37-s − 1.87·41-s − 1.21·43-s + 0.298·45-s + 0.144·48-s + 1/7·49-s − 1.68·51-s − 3.12·59-s + 0.258·60-s + 0.125·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 926100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.988751310608497948309425656861, −7.48230039465939347417283743898, −6.88089320758720461322150711491, −6.63010526050831562050636982417, −6.43968643454495280602009146366, −5.75531801300200717416192667829, −4.97103303999261607614028493172, −4.92567559830316004423556337525, −4.33770231717830777250549003341, −3.45283173381757043345013171615, −3.22977772739188491587716942834, −2.33971670719236581491979893835, −1.84101162653270326557570577442, −1.69594451483764738863265937021, 0,
1.69594451483764738863265937021, 1.84101162653270326557570577442, 2.33971670719236581491979893835, 3.22977772739188491587716942834, 3.45283173381757043345013171615, 4.33770231717830777250549003341, 4.92567559830316004423556337525, 4.97103303999261607614028493172, 5.75531801300200717416192667829, 6.43968643454495280602009146366, 6.63010526050831562050636982417, 6.88089320758720461322150711491, 7.48230039465939347417283743898, 7.988751310608497948309425656861
