| L(s) = 1 | + 4·9-s + 104·29-s + 1.36e3·41-s + 1.26e3·49-s − 1.04e3·61-s − 1.44e3·81-s + 2.52e3·89-s + 6.55e3·101-s + 1.36e3·109-s − 1.58e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.47e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4/27·9-s + 0.665·29-s + 5.21·41-s + 3.69·49-s − 2.19·61-s − 1.98·81-s + 3.00·89-s + 6.45·101-s + 1.20·109-s − 1.18·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.94·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.633588122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.633588122\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 634 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 790 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3238 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 3170 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 19398 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 49390 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 78806 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 342 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 47374 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 133526 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 229110 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 170310 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 353954 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 392610 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 312910 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 945310 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 1120642 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 892190 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−7.19590224590867659844842486979, −6.56790798161080948142092437747, −6.50803207029225268712066845463, −6.11552646932547076415093125880, −6.03589780721043224458436720510, −5.95350985926991549707500484809, −5.66602121809587128829935309799, −5.22702807906520805963327621027, −5.19059664545849985676010933386, −4.73244565037062254835258461224, −4.39149373134623777934327201002, −4.30495715744782972816984553624, −4.28815604580371402362190468848, −3.78768803799983053548539296967, −3.54692558895123923305297859825, −3.10299030172198125957693878967, −3.05730898651425979489855334745, −2.50245042212019882946339593310, −2.33389482330051400440407788869, −2.19082098171038456967365281350, −1.78697655525800948989094874240, −1.04470883928351224473879465820, −1.03848011832632045960713317062, −0.71381845431506083813645043350, −0.35838593777487912805694782091,
0.35838593777487912805694782091, 0.71381845431506083813645043350, 1.03848011832632045960713317062, 1.04470883928351224473879465820, 1.78697655525800948989094874240, 2.19082098171038456967365281350, 2.33389482330051400440407788869, 2.50245042212019882946339593310, 3.05730898651425979489855334745, 3.10299030172198125957693878967, 3.54692558895123923305297859825, 3.78768803799983053548539296967, 4.28815604580371402362190468848, 4.30495715744782972816984553624, 4.39149373134623777934327201002, 4.73244565037062254835258461224, 5.19059664545849985676010933386, 5.22702807906520805963327621027, 5.66602121809587128829935309799, 5.95350985926991549707500484809, 6.03589780721043224458436720510, 6.11552646932547076415093125880, 6.50803207029225268712066845463, 6.56790798161080948142092437747, 7.19590224590867659844842486979
