| L(s) = 1 | + 4·3-s + 6·9-s + 12·11-s − 12·17-s + 4·19-s − 10·25-s − 4·27-s + 48·33-s + 12·41-s − 20·43-s − 14·49-s − 48·51-s + 16·57-s + 12·59-s − 28·67-s − 4·73-s − 40·75-s − 37·81-s + 36·83-s − 36·89-s + 20·97-s + 72·99-s + 12·107-s + 36·113-s + 86·121-s + 48·123-s + 127-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s + 3.61·11-s − 2.91·17-s + 0.917·19-s − 2·25-s − 0.769·27-s + 8.35·33-s + 1.87·41-s − 3.04·43-s − 2·49-s − 6.72·51-s + 2.11·57-s + 1.56·59-s − 3.42·67-s − 0.468·73-s − 4.61·75-s − 4.11·81-s + 3.95·83-s − 3.81·89-s + 2.03·97-s + 7.23·99-s + 1.16·107-s + 3.38·113-s + 7.81·121-s + 4.32·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.172496733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.172496733\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.63246815326574922262150441656, −13.87802765345646590538282788523, −13.87802765345646590538282788523, −13.21770084467032340011509610388, −13.21770084467032340011509610388, −11.91995920497978496826321746442, −11.91995920497978496826321746442, −11.22420990477091317474434780207, −11.22420990477091317474434780207, −9.739977345448053506810012798737, −9.739977345448053506810012798737, −9.084650424863502138877874475320, −9.084650424863502138877874475320, −8.337928305679105143801605101388, −8.337928305679105143801605101388, −7.14838005963241329022744903293, −7.14838005963241329022744903293, −6.13287781737238367476060833635, −6.13287781737238367476060833635, −4.38680180444175422137402538929, −4.38680180444175422137402538929, −3.38746738060370983088123271545, −3.38746738060370983088123271545, −1.90996633779170653420430529631, −1.90996633779170653420430529631,
1.90996633779170653420430529631, 1.90996633779170653420430529631, 3.38746738060370983088123271545, 3.38746738060370983088123271545, 4.38680180444175422137402538929, 4.38680180444175422137402538929, 6.13287781737238367476060833635, 6.13287781737238367476060833635, 7.14838005963241329022744903293, 7.14838005963241329022744903293, 8.337928305679105143801605101388, 8.337928305679105143801605101388, 9.084650424863502138877874475320, 9.084650424863502138877874475320, 9.739977345448053506810012798737, 9.739977345448053506810012798737, 11.22420990477091317474434780207, 11.22420990477091317474434780207, 11.91995920497978496826321746442, 11.91995920497978496826321746442, 13.21770084467032340011509610388, 13.21770084467032340011509610388, 13.87802765345646590538282788523, 13.87802765345646590538282788523, 14.63246815326574922262150441656
