| L(s) = 1 | − 11·4-s + 57·16-s − 328·19-s − 464·31-s − 686·49-s − 716·61-s + 77·64-s + 3.60e3·76-s + 608·79-s + 3.66e3·109-s − 2.66e3·121-s + 5.10e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.37·4-s + 0.890·16-s − 3.96·19-s − 2.68·31-s − 2·49-s − 1.50·61-s + 0.150·64-s + 5.44·76-s + 0.865·79-s + 3.22·109-s − 2·121-s + 3.69·124-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 − 2·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 11 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9394 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 14654 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 232 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90034 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 88666 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 358 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 304 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 469546 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
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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
−11.38384544889922558773300890374, −11.01686248198233883637036065022, −10.55833654057932124804499891219, −10.17634217525182380976914238922, −9.405917369619758315077402180390, −9.092320934900559766495102421100, −8.621951082404838338011289046138, −8.312744867927485936450696056011, −7.67121288089239657181533416191, −6.97541723465667528651000855303, −6.23992089177508620461553919967, −6.00312121854706366471149544898, −4.92612961970461821363881523543, −4.74787374798103495310982545876, −3.93653285319028387175081764297, −3.64880512614452624507333555275, −2.36746407613640204428682890090, −1.67563667363456175278017021165, 0, 0,
1.67563667363456175278017021165, 2.36746407613640204428682890090, 3.64880512614452624507333555275, 3.93653285319028387175081764297, 4.74787374798103495310982545876, 4.92612961970461821363881523543, 6.00312121854706366471149544898, 6.23992089177508620461553919967, 6.97541723465667528651000855303, 7.67121288089239657181533416191, 8.312744867927485936450696056011, 8.621951082404838338011289046138, 9.092320934900559766495102421100, 9.405917369619758315077402180390, 10.17634217525182380976914238922, 10.55833654057932124804499891219, 11.01686248198233883637036065022, 11.38384544889922558773300890374
