| L(s) = 1 | − 8·2-s + 18·3-s + 48·4-s − 42·5-s − 144·6-s − 98·7-s − 256·8-s + 243·9-s + 336·10-s + 242·11-s + 864·12-s − 230·13-s + 784·14-s − 756·15-s + 1.28e3·16-s − 1.22e3·17-s − 1.94e3·18-s + 1.33e3·19-s − 2.01e3·20-s − 1.76e3·21-s − 1.93e3·22-s + 2.18e3·23-s − 4.60e3·24-s − 2.12e3·25-s + 1.84e3·26-s + 2.91e3·27-s − 4.70e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.751·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.06·10-s + 0.603·11-s + 1.73·12-s − 0.377·13-s + 1.06·14-s − 0.867·15-s + 5/4·16-s − 1.02·17-s − 1.41·18-s + 0.845·19-s − 1.12·20-s − 0.872·21-s − 0.852·22-s + 0.860·23-s − 1.63·24-s − 0.680·25-s + 0.533·26-s + 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 42 T + 3891 T^{2} + 42 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 230 T + 706419 T^{2} + 230 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1220 T + 2716522 T^{2} + 1220 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 70 p T + 3428123 T^{2} - 70 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2184 T + 9517698 T^{2} - 2184 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8614 T + 58337747 T^{2} - 8614 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1500 T + 630874 p T^{2} - 1500 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 17462 T + 183576083 T^{2} + 17462 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2096 T + 167607778 T^{2} - 2096 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 384 p T + 355206394 T^{2} + 384 p^{6} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 40030 T + 781107491 T^{2} + 40030 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1328 T + 836435374 T^{2} + 1328 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10022 T + 163663651 T^{2} + 10022 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 22004 T + 1182735438 T^{2} - 22004 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 47878 T + 3136199603 T^{2} + 47878 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 41872 T + 3809667998 T^{2} - 41872 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 32702 T + 4272503379 T^{2} - 32702 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 63944 T + 7175848382 T^{2} + 63944 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 74260 T + 8475827278 T^{2} - 74260 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 48980 T + 692967766 T^{2} + 48980 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 198056 T + 26523541526 T^{2} + 198056 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.719887224030528351762323001853, −9.625254327574147287225621045624, −9.104430947602087190975192764032, −8.632395891685066084816763369397, −8.243922622903058327156417633407, −8.080919280549510181188252298429, −7.18216566548328119661749035766, −7.14207313461383886844384509175, −6.50800377764117320383020989900, −6.31092301384877671336989506973, −5.08776185318114486135723649037, −4.79668665915188286531372037127, −3.66745517762462209214728668234, −3.63916497210880088837374324554, −2.81028893982998601327268135864, −2.50137646047230040908859400890, −1.51514393513244234296976199632, −1.21562189264899526818385551565, 0, 0,
1.21562189264899526818385551565, 1.51514393513244234296976199632, 2.50137646047230040908859400890, 2.81028893982998601327268135864, 3.63916497210880088837374324554, 3.66745517762462209214728668234, 4.79668665915188286531372037127, 5.08776185318114486135723649037, 6.31092301384877671336989506973, 6.50800377764117320383020989900, 7.14207313461383886844384509175, 7.18216566548328119661749035766, 8.080919280549510181188252298429, 8.243922622903058327156417633407, 8.632395891685066084816763369397, 9.104430947602087190975192764032, 9.625254327574147287225621045624, 9.719887224030528351762323001853
