| L(s) = 1 | − 4-s − 5-s − 7-s − 11-s + 2·17-s − 19-s + 20-s − 23-s + 25-s + 28-s + 35-s − 2·43-s + 44-s − 47-s + 55-s + 61-s + 64-s − 2·68-s + 73-s + 76-s + 77-s + 2·83-s − 2·85-s + 92-s + 95-s − 100-s − 101-s + ⋯ |
| L(s) = 1 | − 4-s − 5-s − 7-s − 11-s + 2·17-s − 19-s + 20-s − 23-s + 25-s + 28-s + 35-s − 2·43-s + 44-s − 47-s + 55-s + 61-s + 64-s − 2·68-s + 73-s + 76-s + 77-s + 2·83-s − 2·85-s + 92-s + 95-s − 100-s − 101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3460192051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3460192051\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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
−10.20122895996263241719723628048, −9.664364677873167465737265116337, −9.495601646962698503901438235066, −8.886718941499348788142839015224, −8.414291248525134310145446427205, −8.050443602837067140613747193959, −7.961850688137002286371386339134, −7.40727355798287391237042119844, −6.71527026784532871597376967832, −6.57783346399589437624763466321, −5.93363931353494773899279051997, −5.36233441454952699206560938217, −5.03163515421800266697213775429, −4.62988585448900648774343310221, −3.88492006267903032827196732962, −3.68220336818178951142601099406, −3.19161647455166701595024768813, −2.62909557852199254758791572345, −1.74621296196262756982297561400, −0.52373332650057176174953844588,
0.52373332650057176174953844588, 1.74621296196262756982297561400, 2.62909557852199254758791572345, 3.19161647455166701595024768813, 3.68220336818178951142601099406, 3.88492006267903032827196732962, 4.62988585448900648774343310221, 5.03163515421800266697213775429, 5.36233441454952699206560938217, 5.93363931353494773899279051997, 6.57783346399589437624763466321, 6.71527026784532871597376967832, 7.40727355798287391237042119844, 7.961850688137002286371386339134, 8.050443602837067140613747193959, 8.414291248525134310145446427205, 8.886718941499348788142839015224, 9.495601646962698503901438235066, 9.664364677873167465737265116337, 10.20122895996263241719723628048
