| L(s) = 1 | − 3·4-s − 3·5-s + 9-s + 5·16-s + 12·19-s + 9·20-s + 4·25-s − 3·36-s − 18·41-s − 3·45-s − 5·49-s − 10·61-s − 3·64-s − 36·76-s − 15·80-s + 81-s − 36·95-s − 12·100-s + 22·109-s − 3·121-s + 3·125-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 149-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 1.34·5-s + 1/3·9-s + 5/4·16-s + 2.75·19-s + 2.01·20-s + 4/5·25-s − 1/2·36-s − 2.81·41-s − 0.447·45-s − 5/7·49-s − 1.28·61-s − 3/8·64-s − 4.12·76-s − 1.67·80-s + 1/9·81-s − 3.69·95-s − 6/5·100-s + 2.10·109-s − 0.272·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 61 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 149 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.019705601658162368313782930300, −7.59525541153033948022717953496, −7.32797799246574372614296143278, −6.82355261140588630948200105590, −6.19665770243692771062088275851, −5.48372292315643443082616494394, −5.10014833933526035500158006766, −4.80604409903523102818660120535, −4.31083468335026787397617500846, −3.67127981302010446405898169929, −3.39753239660558066629224529274, −2.97813766484331641600779239042, −1.67112093006873605836011024535, −0.903557978499670931978921105460, 0,
0.903557978499670931978921105460, 1.67112093006873605836011024535, 2.97813766484331641600779239042, 3.39753239660558066629224529274, 3.67127981302010446405898169929, 4.31083468335026787397617500846, 4.80604409903523102818660120535, 5.10014833933526035500158006766, 5.48372292315643443082616494394, 6.19665770243692771062088275851, 6.82355261140588630948200105590, 7.32797799246574372614296143278, 7.59525541153033948022717953496, 8.019705601658162368313782930300
