| L(s) = 1 | + 5-s + 2·7-s − 9-s − 2·11-s + 17-s − 19-s + 23-s + 25-s + 2·35-s − 2·43-s − 45-s − 2·47-s + 3·49-s − 2·55-s − 2·61-s − 2·63-s − 2·73-s − 4·77-s − 2·83-s + 85-s − 95-s + 2·99-s + 101-s + 115-s + 2·119-s + 121-s + 2·125-s + ⋯ |
| L(s) = 1 | + 5-s + 2·7-s − 9-s − 2·11-s + 17-s − 19-s + 23-s + 25-s + 2·35-s − 2·43-s − 45-s − 2·47-s + 3·49-s − 2·55-s − 2·61-s − 2·63-s − 2·73-s − 4·77-s − 2·83-s + 85-s − 95-s + 2·99-s + 101-s + 115-s + 2·119-s + 121-s + 2·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9551621727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9551621727\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−11.13959054975195862916308810271, −10.80586181852080632349541324149, −10.43417693083198184373789194958, −10.16644384465903336371900381510, −9.564291806996441187246636934588, −8.918145196570350113020173579221, −8.431050937786117325797763253107, −8.297768829904437201319863794593, −7.84483887038179014517738524124, −7.33882750159943203159656852194, −6.77891612716172954239535247220, −6.02470916889726814118017157017, −5.44669032102847998337813284477, −5.43626596634302760288955360460, −4.66490704319140395632118686786, −4.57714236045013349233443617036, −3.07235717316408020694215280717, −2.97858536469963990867659022587, −2.02971190226874062559952940153, −1.54675859337263129998469221624,
1.54675859337263129998469221624, 2.02971190226874062559952940153, 2.97858536469963990867659022587, 3.07235717316408020694215280717, 4.57714236045013349233443617036, 4.66490704319140395632118686786, 5.43626596634302760288955360460, 5.44669032102847998337813284477, 6.02470916889726814118017157017, 6.77891612716172954239535247220, 7.33882750159943203159656852194, 7.84483887038179014517738524124, 8.297768829904437201319863794593, 8.431050937786117325797763253107, 8.918145196570350113020173579221, 9.564291806996441187246636934588, 10.16644384465903336371900381510, 10.43417693083198184373789194958, 10.80586181852080632349541324149, 11.13959054975195862916308810271
