| L(s) = 1 | − 4·5-s − 4·17-s + 6·25-s − 12·29-s − 8·37-s + 12·41-s + 6·49-s + 4·53-s + 8·61-s + 12·73-s + 16·85-s + 12·89-s − 12·97-s + 20·101-s + 16·109-s + 12·113-s − 6·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.970·17-s + 6/5·25-s − 2.22·29-s − 1.31·37-s + 1.87·41-s + 6/7·49-s + 0.549·53-s + 1.02·61-s + 1.40·73-s + 1.73·85-s + 1.27·89-s − 1.21·97-s + 1.99·101-s + 1.53·109-s + 1.12·113-s − 0.545·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.52571339957459172637957532876, −7.43883959737666870681043698835, −7.14983127219675626592699180132, −6.53629486520291897182011452030, −5.99546954745022357178000891610, −5.55450010906413996473647644643, −4.98661739989470548034538066281, −4.50359273326774206201263933145, −3.97896514146790989741610882364, −3.71401255404716107940639850583, −3.31634970464710210538375237259, −2.38684579798418103670547927964, −1.98522850455648645194397487369, −0.811401471347739942097340786558, 0,
0.811401471347739942097340786558, 1.98522850455648645194397487369, 2.38684579798418103670547927964, 3.31634970464710210538375237259, 3.71401255404716107940639850583, 3.97896514146790989741610882364, 4.50359273326774206201263933145, 4.98661739989470548034538066281, 5.55450010906413996473647644643, 5.99546954745022357178000891610, 6.53629486520291897182011452030, 7.14983127219675626592699180132, 7.43883959737666870681043698835, 7.52571339957459172637957532876
