| L(s) = 1 | − 4·2-s + 8·3-s + 10·5-s − 32·6-s + 64·8-s + 243·9-s − 40·10-s + 340·11-s + 588·13-s + 80·15-s − 256·16-s + 1.22e3·17-s − 972·18-s + 2.43e3·19-s − 1.36e3·22-s − 2.00e3·23-s + 512·24-s + 3.12e3·25-s − 2.35e3·26-s + 5.32e3·27-s − 1.34e4·29-s − 320·30-s + 8.85e3·31-s + 2.72e3·33-s − 4.90e3·34-s − 9.18e3·37-s − 9.72e3·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.513·3-s + 0.178·5-s − 0.362·6-s + 0.353·8-s + 9-s − 0.126·10-s + 0.847·11-s + 0.964·13-s + 0.0918·15-s − 1/4·16-s + 1.02·17-s − 0.707·18-s + 1.54·19-s − 0.599·22-s − 0.788·23-s + 0.181·24-s + 25-s − 0.682·26-s + 1.40·27-s − 2.97·29-s − 0.0649·30-s + 1.65·31-s + 0.434·33-s − 0.727·34-s − 1.10·37-s − 1.09·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.115155571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.115155571\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 8 T - 179 T^{2} - 8 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 p T - 121 p^{2} T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 340 T - 45451 T^{2} - 340 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 294 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1226 T + 83219 T^{2} - 1226 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 128 p T + 9525 p^{2} T^{2} - 128 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2000 T - 2436343 T^{2} + 2000 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6746 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8856 T + 49799585 T^{2} - 8856 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 9182 T + 14965167 T^{2} + 9182 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 14574 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8108 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 312 T - 229247663 T^{2} + 312 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14634 T - 204041537 T^{2} - 14634 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 27656 T + 49930037 T^{2} + 27656 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 34338 T + 334501943 T^{2} - 34338 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12316 T - 1198441251 T^{2} + 12316 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 520 p T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 61718 T + 1736039931 T^{2} + 61718 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 64752 T + 1115765105 T^{2} - 64752 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 77056 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8166 T - 5517375893 T^{2} + 8166 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 20650 T + p^{5} T^{2} )^{2} \) |
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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
−13.33167229204887440184165386781, −12.67512342289275162792482786010, −12.18977843136608508826676517796, −11.65514350389488905911826455548, −10.79564491667310531605785523060, −10.61289694244564348629076968031, −9.580235374062774738262237550961, −9.556760787207023670902264270398, −9.004362313968316890369180351948, −8.302461948368666604670333827220, −7.56544129747300337121846496260, −7.39097574345423083230729532353, −6.45343580312168920856036236687, −5.79486229727359108365018410088, −4.99533068966914636599288814763, −3.95722977520276445328771021313, −3.59433693640265753009694177848, −2.44248386899417098885041370125, −1.26790760072664839952306218185, −0.951773173265855325772957131715,
0.951773173265855325772957131715, 1.26790760072664839952306218185, 2.44248386899417098885041370125, 3.59433693640265753009694177848, 3.95722977520276445328771021313, 4.99533068966914636599288814763, 5.79486229727359108365018410088, 6.45343580312168920856036236687, 7.39097574345423083230729532353, 7.56544129747300337121846496260, 8.302461948368666604670333827220, 9.004362313968316890369180351948, 9.556760787207023670902264270398, 9.580235374062774738262237550961, 10.61289694244564348629076968031, 10.79564491667310531605785523060, 11.65514350389488905911826455548, 12.18977843136608508826676517796, 12.67512342289275162792482786010, 13.33167229204887440184165386781
