| L(s) = 1 | − 512·4-s − 289·5-s − 1.31e4·9-s − 5.00e4·11-s + 1.96e5·16-s + 2.60e5·19-s + 1.47e5·20-s − 3.07e5·25-s + 6.71e6·36-s + 2.56e7·44-s + 3.79e6·45-s − 1.12e7·49-s + 1.44e7·55-s + 3.53e7·61-s − 6.71e7·64-s − 1.33e8·76-s − 5.68e7·80-s + 1.29e8·81-s − 7.53e7·95-s + 6.56e8·99-s + 1.57e8·100-s + 2.16e8·101-s + 1.44e9·121-s + 2.01e8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.462·5-s − 2·9-s − 3.41·11-s + 3·16-s + 2·19-s + 0.924·20-s − 0.786·25-s + 4·36-s + 6.83·44-s + 0.924·45-s − 1.95·49-s + 1.57·55-s + 2.55·61-s − 4·64-s − 4·76-s − 1.38·80-s + 3·81-s − 0.924·95-s + 6.83·99-s + 1.57·100-s + 2.07·101-s + 6.75·121-s + 0.825·125-s − 6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2693572975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2693572975\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 + 289 T + p^{8} T^{2} \) |
| 19 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 527 T + p^{8} T^{2} )( 1 + 527 T + p^{8} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 25007 T + p^{8} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 42433 T + p^{8} T^{2} )( 1 + 42433 T + p^{8} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 534718 T + p^{8} T^{2} )( 1 + 534718 T + p^{8} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5602127 T + p^{8} T^{2} )( 1 + 5602127 T + p^{8} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8302513 T + p^{8} T^{2} )( 1 + 8302513 T + p^{8} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 17661793 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 43864607 T + p^{8} T^{2} )( 1 + 43864607 T + p^{8} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 62676958 T + p^{8} T^{2} )( 1 + 62676958 T + p^{8} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
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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
−13.06482801214118280544276080641, −12.14200549126359614842940250795, −11.61349287628487016043739062860, −11.00049870000897647136142706365, −10.38395981365101916998289684155, −9.841641951633655390877932139501, −9.441254722039633835302120698325, −8.598847084065808033360108709647, −8.157847498349727484338907977679, −7.966856900047827985253232132310, −7.44001629790758694233100463507, −5.88088172116648961382857703210, −5.35254455010733943196390208267, −5.27362088235628917902224698127, −4.62878272543920770581143340095, −3.35738187366264282516552921442, −3.20982920975004536173935117487, −2.35716366250492988927084851065, −0.69765245666182102899342331833, −0.24747955139998980848507544441,
0.24747955139998980848507544441, 0.69765245666182102899342331833, 2.35716366250492988927084851065, 3.20982920975004536173935117487, 3.35738187366264282516552921442, 4.62878272543920770581143340095, 5.27362088235628917902224698127, 5.35254455010733943196390208267, 5.88088172116648961382857703210, 7.44001629790758694233100463507, 7.966856900047827985253232132310, 8.157847498349727484338907977679, 8.598847084065808033360108709647, 9.441254722039633835302120698325, 9.841641951633655390877932139501, 10.38395981365101916998289684155, 11.00049870000897647136142706365, 11.61349287628487016043739062860, 12.14200549126359614842940250795, 13.06482801214118280544276080641
