| L(s) = 1 | − 5-s + 4·7-s − 11-s − 6·13-s − 2·17-s + 5·19-s + 7·23-s + 12·29-s − 4·31-s − 4·35-s + 5·37-s + 10·41-s + 12·43-s − 9·47-s + 9·49-s + 11·53-s + 55-s + 8·59-s + 12·61-s + 6·65-s + 4·67-s + 8·71-s − 12·73-s − 4·77-s − 14·79-s + 8·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s − 1.66·13-s − 0.485·17-s + 1.14·19-s + 1.45·23-s + 2.22·29-s − 0.718·31-s − 0.676·35-s + 0.821·37-s + 1.56·41-s + 1.82·43-s − 1.31·47-s + 9/7·49-s + 1.51·53-s + 0.134·55-s + 1.04·59-s + 1.53·61-s + 0.744·65-s + 0.488·67-s + 0.949·71-s − 1.40·73-s − 0.455·77-s − 1.57·79-s + 0.878·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.236586461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.236586461\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.972297904479883238945647392118, −8.791946621318299684379746035300, −8.119280721857280814751684692802, −8.068892527288288142877442889650, −7.44288011746222660204987632086, −7.42581049719476043395770205293, −6.85275071721364271345223966334, −6.67278328980863737766427790831, −5.72948531995675413185595193614, −5.60254243951814978616321989377, −4.98385438459882192804490140952, −4.89492322202028181960637548424, −4.28800219002226862995000045631, −4.20051254021827887462923809206, −3.30838618782954384664330983332, −2.79531972754926024125097723619, −2.44757560121531930078913341989, −2.00001611572601599166939260611, −0.980643232192021474666304123604, −0.77128339469304737573022260051,
0.77128339469304737573022260051, 0.980643232192021474666304123604, 2.00001611572601599166939260611, 2.44757560121531930078913341989, 2.79531972754926024125097723619, 3.30838618782954384664330983332, 4.20051254021827887462923809206, 4.28800219002226862995000045631, 4.89492322202028181960637548424, 4.98385438459882192804490140952, 5.60254243951814978616321989377, 5.72948531995675413185595193614, 6.67278328980863737766427790831, 6.85275071721364271345223966334, 7.42581049719476043395770205293, 7.44288011746222660204987632086, 8.068892527288288142877442889650, 8.119280721857280814751684692802, 8.791946621318299684379746035300, 8.972297904479883238945647392118
