| L(s) = 1 | − 2-s − 3·3-s − 5-s + 3·6-s + 2·7-s + 8-s + 6·9-s + 10-s − 4·11-s − 7·13-s − 2·14-s + 3·15-s − 16-s − 7·17-s − 6·18-s − 19-s − 6·21-s + 4·22-s + 6·23-s − 3·24-s + 5·25-s + 7·26-s − 9·27-s − 6·29-s − 3·30-s + 7·31-s + 12·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 0.447·5-s + 1.22·6-s + 0.755·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 1.20·11-s − 1.94·13-s − 0.534·14-s + 0.774·15-s − 1/4·16-s − 1.69·17-s − 1.41·18-s − 0.229·19-s − 1.30·21-s + 0.852·22-s + 1.25·23-s − 0.612·24-s + 25-s + 1.37·26-s − 1.73·27-s − 1.11·29-s − 0.547·30-s + 1.25·31-s + 2.08·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2504835501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2504835501\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−12.62950336776362834288227477311, −11.54384456155282791138014500674, −11.41921024566657346546370159419, −11.06118567233736727189840807078, −10.64559148883535576790675336170, −10.02763807828278873677062399261, −9.738516245441740608924730830309, −9.127621579876917035966212246415, −8.294237611889375955705362533278, −8.063066597226526063443247626508, −7.34149075361801390177133548258, −6.78377365433153306261818581578, −6.70635440531745501244315141001, −5.42447070190230526907917088766, −5.24356280733702663316269759244, −4.53130058097212535294355118558, −4.45669170973107871790583799204, −2.86502920266357870826712876930, −1.97517693997973803964703024685, −0.50158623033298012806731316187,
0.50158623033298012806731316187, 1.97517693997973803964703024685, 2.86502920266357870826712876930, 4.45669170973107871790583799204, 4.53130058097212535294355118558, 5.24356280733702663316269759244, 5.42447070190230526907917088766, 6.70635440531745501244315141001, 6.78377365433153306261818581578, 7.34149075361801390177133548258, 8.063066597226526063443247626508, 8.294237611889375955705362533278, 9.127621579876917035966212246415, 9.738516245441740608924730830309, 10.02763807828278873677062399261, 10.64559148883535576790675336170, 11.06118567233736727189840807078, 11.41921024566657346546370159419, 11.54384456155282791138014500674, 12.62950336776362834288227477311
