| L(s) = 1 | − 4-s + 2·5-s − 4·9-s − 3·13-s − 3·16-s − 4·17-s − 3·19-s − 2·20-s − 3·23-s − 2·25-s + 4·36-s − 37-s − 2·43-s − 8·45-s − 47-s + 4·49-s + 3·52-s + 7·64-s − 6·65-s + 4·68-s − 14·71-s + 3·76-s + 10·79-s − 6·80-s + 7·81-s − 8·85-s + 3·92-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s − 4/3·9-s − 0.832·13-s − 3/4·16-s − 0.970·17-s − 0.688·19-s − 0.447·20-s − 0.625·23-s − 2/5·25-s + 2/3·36-s − 0.164·37-s − 0.304·43-s − 1.19·45-s − 0.145·47-s + 4/7·49-s + 0.416·52-s + 7/8·64-s − 0.744·65-s + 0.485·68-s − 1.66·71-s + 0.344·76-s + 1.12·79-s − 0.670·80-s + 7/9·81-s − 0.867·85-s + 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51005 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51005 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 101 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
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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
−9.728700651601606822751472445507, −9.291419600896184591458340969488, −8.813024625233088576583383817327, −8.525522793869626337956143518921, −7.82578318499584537782581818852, −7.20152138759499360168550782180, −6.46335991685157961372547360255, −6.09493305416815985505427508169, −5.51864259033180555366486250251, −4.86439631929958785852821223888, −4.37067453180056733328480562447, −3.47697954271673039561109068255, −2.46604034157258189949375676810, −2.07847157603502276997862052835, 0,
2.07847157603502276997862052835, 2.46604034157258189949375676810, 3.47697954271673039561109068255, 4.37067453180056733328480562447, 4.86439631929958785852821223888, 5.51864259033180555366486250251, 6.09493305416815985505427508169, 6.46335991685157961372547360255, 7.20152138759499360168550782180, 7.82578318499584537782581818852, 8.525522793869626337956143518921, 8.813024625233088576583383817327, 9.291419600896184591458340969488, 9.728700651601606822751472445507
