| L(s) = 1 | − 64·4-s + 1.51e4·11-s + 4.09e3·16-s + 1.02e5·19-s + 9.40e4·29-s − 3.84e5·31-s + 4.75e5·41-s − 9.68e5·44-s + 2.68e5·49-s + 3.00e6·59-s − 4.13e6·61-s − 2.62e5·64-s − 8.24e6·71-s − 6.55e6·76-s − 2.90e6·79-s + 1.20e7·89-s + 7.91e6·101-s + 2.44e7·109-s − 6.02e6·116-s + 1.32e8·121-s + 2.46e7·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 3.42·11-s + 1/4·16-s + 3.42·19-s + 0.716·29-s − 2.31·31-s + 1.07·41-s − 1.71·44-s + 0.326·49-s + 1.90·59-s − 2.33·61-s − 1/8·64-s − 2.73·71-s − 1.71·76-s − 0.663·79-s + 1.80·89-s + 0.764·101-s + 1.80·109-s − 0.358·116-s + 6.80·121-s + 1.15·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.327408102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.327408102\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 268810 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 7563 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 96638650 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 843145 p^{2} T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 51235 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3489816970 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 47040 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 192358 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 151028745910 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 237723 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 117212550070 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 329509091470 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2025636243190 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 1501080 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2068918 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 257883176845 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4121052 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 22087799213785 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1454030 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 51626381773765 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6004335 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 149956558187710 T^{2} + p^{14} T^{4} \) |
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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
−9.923374914748787078762963026552, −9.598125240927485583771544026269, −9.187059666983558371329088601495, −8.923446492308061568783926699954, −8.723000558343305132376587159244, −7.60748434435774759863021983543, −7.39465321681355895676386792606, −7.14600376422219506256592127201, −6.21451142370529435875862578363, −6.18286103332631702848611466223, −5.39665676722029548517522317807, −5.02765887727846932244141304907, −4.08770703395659494188646269971, −4.08569328755746368262881645946, −3.25826170708675744754859853191, −3.16391172735101543961543113472, −1.93555770967744663444818122795, −1.21917058282298741209046473426, −1.21708766344438433398644374452, −0.53036075983103706953905413070,
0.53036075983103706953905413070, 1.21708766344438433398644374452, 1.21917058282298741209046473426, 1.93555770967744663444818122795, 3.16391172735101543961543113472, 3.25826170708675744754859853191, 4.08569328755746368262881645946, 4.08770703395659494188646269971, 5.02765887727846932244141304907, 5.39665676722029548517522317807, 6.18286103332631702848611466223, 6.21451142370529435875862578363, 7.14600376422219506256592127201, 7.39465321681355895676386792606, 7.60748434435774759863021983543, 8.723000558343305132376587159244, 8.923446492308061568783926699954, 9.187059666983558371329088601495, 9.598125240927485583771544026269, 9.923374914748787078762963026552
