| L(s) = 1 | − 128·4-s − 54·5-s − 1.45e3·9-s + 2.12e3·11-s + 1.22e4·16-s − 1.37e4·19-s + 6.91e3·20-s − 1.27e4·25-s + 1.86e5·36-s − 2.71e5·44-s + 7.87e4·45-s + 1.36e5·49-s − 1.14e5·55-s + 1.14e5·61-s − 1.04e6·64-s + 1.75e6·76-s − 6.63e5·80-s + 1.59e6·81-s + 7.40e5·95-s − 3.09e6·99-s + 1.62e6·100-s − 4.12e6·101-s − 1.59e5·121-s + 1.53e6·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.431·5-s − 2·9-s + 1.59·11-s + 3·16-s − 2·19-s + 0.863·20-s − 0.813·25-s + 4·36-s − 3.19·44-s + 0.863·45-s + 1.16·49-s − 0.689·55-s + 0.502·61-s − 4·64-s + 4·76-s − 1.29·80-s + 3·81-s + 0.863·95-s − 3.19·99-s + 1.62·100-s − 3.99·101-s − 0.0900·121-s + 0.783·125-s − 6·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.07737587834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07737587834\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + 54 T + p^{6} T^{2} \) |
| 19 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 610 T + p^{6} T^{2} )( 1 + 610 T + p^{6} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 1062 T + p^{6} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 9630 T + p^{6} T^{2} )( 1 + 9630 T + p^{6} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 20610 T + p^{6} T^{2} )( 1 + 20610 T + p^{6} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 142630 T + p^{6} T^{2} )( 1 + 142630 T + p^{6} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 75150 T + p^{6} T^{2} )( 1 + 75150 T + p^{6} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 57062 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 384050 T + p^{6} T^{2} )( 1 + 384050 T + p^{6} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1131030 T + p^{6} T^{2} )( 1 + 1131030 T + p^{6} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
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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
−13.19927034997158456077694103824, −12.40714377311029730566851134878, −12.20405974351430893336139821545, −11.51465693888797487826532421757, −10.99336194548123138165772051105, −10.30914252991741737083960805858, −9.590349932276628469463582780416, −9.025452411347736902397730797180, −8.749127113796881097990257725744, −8.321283471745709724509240336353, −7.80868610697612101674873429137, −6.63171806772919136295828339271, −5.99335947560944753396884850568, −5.47846067914526059839591439712, −4.65985000252412601600183141484, −3.91407554457576303444282516296, −3.70400221204612739074643283617, −2.52937749526561729408169231161, −1.16575102145797551655946574655, −0.11488636310886792373653368724,
0.11488636310886792373653368724, 1.16575102145797551655946574655, 2.52937749526561729408169231161, 3.70400221204612739074643283617, 3.91407554457576303444282516296, 4.65985000252412601600183141484, 5.47846067914526059839591439712, 5.99335947560944753396884850568, 6.63171806772919136295828339271, 7.80868610697612101674873429137, 8.321283471745709724509240336353, 8.749127113796881097990257725744, 9.025452411347736902397730797180, 9.590349932276628469463582780416, 10.30914252991741737083960805858, 10.99336194548123138165772051105, 11.51465693888797487826532421757, 12.20405974351430893336139821545, 12.40714377311029730566851134878, 13.19927034997158456077694103824
