| L(s) = 1 | + 250·5-s − 478·9-s + 4.68e4·25-s + 8.97e4·29-s + 1.48e5·41-s − 1.19e5·45-s − 3.92e4·49-s − 9.04e5·61-s − 3.02e5·81-s + 1.02e6·89-s − 3.83e6·101-s + 2.59e6·109-s + 3.54e6·121-s + 7.81e6·125-s + 127-s + 131-s + 137-s + 139-s + 2.24e7·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.65e6·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2·5-s − 0.655·9-s + 3·25-s + 3.67·29-s + 2.15·41-s − 1.31·45-s − 0.333·49-s − 3.98·61-s − 0.570·81-s + 1.44·89-s − 3.72·101-s + 2.00·109-s + 2·121-s + 4·125-s + 7.35·145-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.166748785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.166748785\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 478 T^{2} + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 39278 T^{2} + p^{12} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 60265042 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 44858 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 74338 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12339826882 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 20859145522 T^{2} + p^{12} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 452342 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 180915136162 T^{2} + p^{12} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 626900357918 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 511058 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
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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
−13.63612836316626212921820111374, −13.01048279665920192598150065655, −12.20569558541493239742325586298, −12.19877475428410839337510895457, −10.86440856072496515021799156451, −10.82344672273167588035113517845, −10.02014079543454939201097337874, −9.653963528210212031753230058190, −8.937751980699468161386846845627, −8.590763824562223588540769869783, −7.74954023717882581770579779836, −6.81494137982628307905060916593, −6.11465105072061553661785771934, −6.00550606604813378248575124742, −4.98594331819744587865480525588, −4.51790689351224735875931547802, −2.89543139070193200067367089156, −2.69443508166515654306745115909, −1.57998549259861019366434282115, −0.799955754260801075993308443295,
0.799955754260801075993308443295, 1.57998549259861019366434282115, 2.69443508166515654306745115909, 2.89543139070193200067367089156, 4.51790689351224735875931547802, 4.98594331819744587865480525588, 6.00550606604813378248575124742, 6.11465105072061553661785771934, 6.81494137982628307905060916593, 7.74954023717882581770579779836, 8.590763824562223588540769869783, 8.937751980699468161386846845627, 9.653963528210212031753230058190, 10.02014079543454939201097337874, 10.82344672273167588035113517845, 10.86440856072496515021799156451, 12.19877475428410839337510895457, 12.20569558541493239742325586298, 13.01048279665920192598150065655, 13.63612836316626212921820111374
