| L(s) = 1 | + 2-s + 2·4-s + 2·5-s + 7-s + 5·8-s + 2·10-s − 4·11-s + 2·13-s + 14-s + 5·16-s − 12·17-s + 8·19-s + 4·20-s − 4·22-s + 5·25-s + 2·26-s + 2·28-s + 2·29-s + 10·32-s − 12·34-s + 2·35-s + 12·37-s + 8·38-s + 10·40-s − 2·41-s + 4·43-s − 8·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.894·5-s + 0.377·7-s + 1.76·8-s + 0.632·10-s − 1.20·11-s + 0.554·13-s + 0.267·14-s + 5/4·16-s − 2.91·17-s + 1.83·19-s + 0.894·20-s − 0.852·22-s + 25-s + 0.392·26-s + 0.377·28-s + 0.371·29-s + 1.76·32-s − 2.05·34-s + 0.338·35-s + 1.97·37-s + 1.29·38-s + 1.58·40-s − 0.312·41-s + 0.609·43-s − 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.341442324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.341442324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−10.84074841618268856319341904131, −10.83870687062690333794455851230, −10.24114864962297505256108136829, −9.754857665629567889206330657246, −9.241014217369785220805842463710, −8.824873028144178037601522614499, −8.040506943750376103797078473193, −7.955310243384355410121115245696, −7.15649449979048994825041226084, −6.94030994085414357730463805521, −6.43708276004130141698751234180, −5.85502430414251615722768096012, −5.39888956364057890435008267544, −4.90544906398926586340458564750, −4.41608069331050401214766423545, −4.03877288359090136441246596474, −2.75938675351521499989349091557, −2.70327867457871969004472983074, −1.94369059287479573153554866939, −1.18238002414778718699540612194,
1.18238002414778718699540612194, 1.94369059287479573153554866939, 2.70327867457871969004472983074, 2.75938675351521499989349091557, 4.03877288359090136441246596474, 4.41608069331050401214766423545, 4.90544906398926586340458564750, 5.39888956364057890435008267544, 5.85502430414251615722768096012, 6.43708276004130141698751234180, 6.94030994085414357730463805521, 7.15649449979048994825041226084, 7.955310243384355410121115245696, 8.040506943750376103797078473193, 8.824873028144178037601522614499, 9.241014217369785220805842463710, 9.754857665629567889206330657246, 10.24114864962297505256108136829, 10.83870687062690333794455851230, 10.84074841618268856319341904131
