| L(s) = 1 | + 2·9-s + 20·41-s − 12·49-s − 40·61-s − 15·81-s + 20·89-s + 8·101-s + 24·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 3.12·41-s − 1.71·49-s − 5.12·61-s − 5/3·81-s + 2.11·89-s + 0.796·101-s + 2.29·109-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.975241442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.975241442\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 129 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.50243333512293403461219937792, −6.47307415905384557449710263302, −6.14792274073187160569514412968, −6.11340335125669985650903256319, −6.11321347121725484928627928177, −5.58089396511053415235143365638, −5.53658995140749345259213975322, −4.92032605350189200925364142557, −4.91783058256341764632572416817, −4.79765888759538529364059775459, −4.66440526546833807150705917231, −4.14729472541376977666057808528, −4.02235531837103089915007582388, −3.86385685788537873863377961264, −3.67712989420087978140641002181, −3.14529845430438537414972332534, −2.92597694131702203629127479998, −2.82972667499162154331001916179, −2.60154891771284060930980880101, −2.15805784182267854612970113496, −1.72217366544716437739480150678, −1.47694149831091236628371696040, −1.43934083527078678654758243277, −0.71340735400597702737733483821, −0.38791313659385024695453233007,
0.38791313659385024695453233007, 0.71340735400597702737733483821, 1.43934083527078678654758243277, 1.47694149831091236628371696040, 1.72217366544716437739480150678, 2.15805784182267854612970113496, 2.60154891771284060930980880101, 2.82972667499162154331001916179, 2.92597694131702203629127479998, 3.14529845430438537414972332534, 3.67712989420087978140641002181, 3.86385685788537873863377961264, 4.02235531837103089915007582388, 4.14729472541376977666057808528, 4.66440526546833807150705917231, 4.79765888759538529364059775459, 4.91783058256341764632572416817, 4.92032605350189200925364142557, 5.53658995140749345259213975322, 5.58089396511053415235143365638, 6.11321347121725484928627928177, 6.11340335125669985650903256319, 6.14792274073187160569514412968, 6.47307415905384557449710263302, 6.50243333512293403461219937792
