L(s) = 1 | − 4-s + 3·5-s + 4·7-s + 2·8-s + 2·9-s + 16-s + 4·17-s − 4·19-s − 3·20-s + 2·25-s − 4·28-s + 4·29-s − 4·32-s + 12·35-s − 2·36-s + 6·40-s + 6·45-s + 9·49-s + 8·56-s − 16·61-s + 8·63-s + 3·64-s − 4·68-s − 8·71-s + 4·72-s + 4·76-s + 3·80-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.34·5-s + 1.51·7-s + 0.707·8-s + 2/3·9-s + 1/4·16-s + 0.970·17-s − 0.917·19-s − 0.670·20-s + 2/5·25-s − 0.755·28-s + 0.742·29-s − 0.707·32-s + 2.02·35-s − 1/3·36-s + 0.948·40-s + 0.894·45-s + 9/7·49-s + 1.06·56-s − 2.04·61-s + 1.00·63-s + 3/8·64-s − 0.485·68-s − 0.949·71-s + 0.471·72-s + 0.458·76-s + 0.335·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259210 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259210 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.880454814\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.880454814\) |
\(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.003996453010516973749175050345, −8.409557068919345985083490412409, −7.968711377540667104722136899585, −7.67077179096122287248501427134, −7.07487060223471216356632464911, −6.52419101923396765956870058105, −5.85261600365733893673759035006, −5.54706514921931500358338861063, −4.91995381140261617866452338232, −4.53115531701632005814600622931, −4.14137260542990120514985312751, −3.23537532507718060033011858467, −2.29716961846542830978715719161, −1.68816949144364586088050169194, −1.23118698247060870292298297224,
1.23118698247060870292298297224, 1.68816949144364586088050169194, 2.29716961846542830978715719161, 3.23537532507718060033011858467, 4.14137260542990120514985312751, 4.53115531701632005814600622931, 4.91995381140261617866452338232, 5.54706514921931500358338861063, 5.85261600365733893673759035006, 6.52419101923396765956870058105, 7.07487060223471216356632464911, 7.67077179096122287248501427134, 7.968711377540667104722136899585, 8.409557068919345985083490412409, 9.003996453010516973749175050345