| L(s) = 1 | − 4·2-s + 8·4-s + 22·5-s − 88·10-s − 110·13-s − 64·16-s + 198·17-s + 176·20-s + 359·25-s + 440·26-s + 256·32-s − 792·34-s + 610·37-s + 460·41-s − 1.43e3·50-s − 880·52-s − 54·53-s + 936·61-s − 512·64-s − 2.42e3·65-s + 1.58e3·68-s − 1.69e3·73-s − 2.44e3·74-s − 1.40e3·80-s − 1.84e3·82-s + 4.35e3·85-s − 2.41e3·97-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.96·5-s − 2.78·10-s − 2.34·13-s − 16-s + 2.82·17-s + 1.96·20-s + 2.87·25-s + 3.31·26-s + 1.41·32-s − 3.99·34-s + 2.71·37-s + 1.75·41-s − 4.06·50-s − 2.34·52-s − 0.139·53-s + 1.96·61-s − 64-s − 4.61·65-s + 2.82·68-s − 2.70·73-s − 3.83·74-s − 1.96·80-s − 2.47·82-s + 5.55·85-s − 2.52·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.695076036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695076036\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 22 T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )( 1 + 92 T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 104 T + p^{3} T^{2} )( 1 - 94 T + p^{3} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 284 T + p^{3} T^{2} )( 1 + 284 T + p^{3} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 - 214 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 518 T + p^{3} T^{2} )( 1 + 572 T + p^{3} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 468 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 592 T + p^{3} T^{2} )( 1 + 1098 T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 176 T + p^{3} T^{2} )( 1 + 176 T + p^{3} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 594 T + p^{3} T^{2} )( 1 + 1816 T + p^{3} T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19197619375912143562727291863, −12.18586479041665735077045643975, −11.16745962359893079119284107019, −10.75312588237723952334037854607, −9.967334571368567288490170931519, −9.845346177619616554737172851300, −9.662425072233978021622129931868, −9.306467683867653924142247336771, −8.381309014347105292121801055766, −7.936852022093572810250836038324, −7.24799083693809827383117378468, −7.09547448469738823196602023091, −6.00456729992620401511077097032, −5.69459062031711880489729599301, −5.07424822499386827882306549765, −4.34900247078349453469326372803, −2.72567720572563008165157217664, −2.59091246844450059936968209483, −1.48771242801720901883568378776, −0.802078665737940531214090709874,
0.802078665737940531214090709874, 1.48771242801720901883568378776, 2.59091246844450059936968209483, 2.72567720572563008165157217664, 4.34900247078349453469326372803, 5.07424822499386827882306549765, 5.69459062031711880489729599301, 6.00456729992620401511077097032, 7.09547448469738823196602023091, 7.24799083693809827383117378468, 7.936852022093572810250836038324, 8.381309014347105292121801055766, 9.306467683867653924142247336771, 9.662425072233978021622129931868, 9.845346177619616554737172851300, 9.967334571368567288490170931519, 10.75312588237723952334037854607, 11.16745962359893079119284107019, 12.18586479041665735077045643975, 12.19197619375912143562727291863
