| L(s) = 1 | + 2·2-s + 3·3-s + 15·5-s + 6·6-s + 35·7-s − 8·8-s + 30·10-s + 9·11-s − 176·13-s + 70·14-s + 45·15-s − 16·16-s + 84·17-s − 104·19-s + 105·21-s + 18·22-s + 84·23-s − 24·24-s + 125·25-s − 352·26-s − 27·27-s + 102·29-s + 90·30-s − 185·31-s + 27·33-s + 168·34-s + 525·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1.34·5-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 0.948·10-s + 0.246·11-s − 3.75·13-s + 1.33·14-s + 0.774·15-s − 1/4·16-s + 1.19·17-s − 1.25·19-s + 1.09·21-s + 0.174·22-s + 0.761·23-s − 0.204·24-s + 25-s − 2.65·26-s − 0.192·27-s + 0.653·29-s + 0.547·30-s − 1.07·31-s + 0.142·33-s + 0.847·34-s + 2.53·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.167316681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.167316681\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 p T + 4 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T - 1250 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 104 T + 3957 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 51 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 185 T + 4434 T^{2} + 185 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 44 T - 48717 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 326 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 138 T - 84779 T^{2} - 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 639 T + 259444 T^{2} + 639 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 159 T - 180098 T^{2} + 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 722 T + 294303 T^{2} + 722 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 166 T - 273207 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1086 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 218 T - 341493 T^{2} + 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 583 T - 153150 T^{2} - 583 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 597 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1038 T + 372475 T^{2} - 1038 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 169 T + p^{3} T^{2} )^{2} \) |
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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
−15.26178417696291290916343252557, −14.94142550864304259034091255737, −14.36934532298448455643604127093, −14.31271419521080606216729321281, −13.90090672773671881444591635387, −12.81602736417602124382330247547, −12.31646414784237188265789769665, −12.07834153877700984898817274917, −10.93250203824558419472067081365, −10.40802741112530223858397834119, −9.408694880818202920241562305098, −9.390360888492254227884243579686, −8.160171991430618535539227517499, −7.60629751575243240440379831064, −6.84023440561201575985433160898, −5.48980522042940511769570272467, −5.07631663123814385231326386770, −4.47319882898785407768263560191, −2.66587029477661293352619596820, −1.97503031701758614170387572369,
1.97503031701758614170387572369, 2.66587029477661293352619596820, 4.47319882898785407768263560191, 5.07631663123814385231326386770, 5.48980522042940511769570272467, 6.84023440561201575985433160898, 7.60629751575243240440379831064, 8.160171991430618535539227517499, 9.390360888492254227884243579686, 9.408694880818202920241562305098, 10.40802741112530223858397834119, 10.93250203824558419472067081365, 12.07834153877700984898817274917, 12.31646414784237188265789769665, 12.81602736417602124382330247547, 13.90090672773671881444591635387, 14.31271419521080606216729321281, 14.36934532298448455643604127093, 14.94142550864304259034091255737, 15.26178417696291290916343252557
