| L(s) = 1 | + 4·2-s + 8·4-s − 18·5-s + 32·8-s − 72·10-s − 50·11-s + 72·13-s + 128·16-s − 126·17-s − 72·19-s − 144·20-s − 200·22-s + 14·23-s + 125·25-s + 288·26-s − 316·29-s − 36·31-s + 256·32-s − 504·34-s + 162·37-s − 288·38-s − 576·40-s − 540·41-s − 648·43-s − 400·44-s + 56·46-s + 72·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.60·5-s + 1.41·8-s − 2.27·10-s − 1.37·11-s + 1.53·13-s + 2·16-s − 1.79·17-s − 0.869·19-s − 1.60·20-s − 1.93·22-s + 0.126·23-s + 25-s + 2.17·26-s − 2.02·29-s − 0.208·31-s + 1.41·32-s − 2.54·34-s + 0.719·37-s − 1.22·38-s − 2.27·40-s − 2.05·41-s − 2.29·43-s − 1.37·44-s + 0.179·46-s + 0.223·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.366469122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366469122\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 18 T + 199 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 50 T + 1169 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 126 T + 10963 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 72 T - 1675 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T - 11971 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 158 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 36 T - 28495 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 162 T - 24409 T^{2} - 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 324 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 72 T - 98639 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 22 T - 148393 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 468 T + 13645 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 792 T + 400283 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 232 T - 246939 T^{2} + 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 734 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 180 T - 356617 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 236 T - 437343 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 234 T - 650213 T^{2} + 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 468 T + p^{3} T^{2} )^{2} \) |
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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
−11.31817281301361835093854035399, −10.81096891988017789577651204555, −10.26585597138870782985597899144, −9.775745831736769278796086303751, −8.841112322935733065054999227772, −8.374405652149641823015832437677, −8.302747422481663395380235616908, −7.44524418860448102189799727865, −7.43475571914835410704903330183, −6.46970666455633211686371445749, −6.40319681149638890623437140234, −5.35301399201040044875213023406, −5.12698689163240792722532058346, −4.56896858250676708856680680352, −3.92239477473000588913223597801, −3.77752386237328314200537615059, −3.23405295333591766424113701887, −2.23427673563790454955284578228, −1.63732336798092830605030125470, −0.28913667217161864760381319130,
0.28913667217161864760381319130, 1.63732336798092830605030125470, 2.23427673563790454955284578228, 3.23405295333591766424113701887, 3.77752386237328314200537615059, 3.92239477473000588913223597801, 4.56896858250676708856680680352, 5.12698689163240792722532058346, 5.35301399201040044875213023406, 6.40319681149638890623437140234, 6.46970666455633211686371445749, 7.43475571914835410704903330183, 7.44524418860448102189799727865, 8.302747422481663395380235616908, 8.374405652149641823015832437677, 8.841112322935733065054999227772, 9.775745831736769278796086303751, 10.26585597138870782985597899144, 10.81096891988017789577651204555, 11.31817281301361835093854035399
