| L(s) = 1 | + 4·2-s + 21·5-s − 74·7-s − 64·8-s + 84·10-s + 270·11-s + 115·13-s − 296·14-s − 256·16-s + 1.72e3·17-s + 3.70e3·19-s + 1.08e3·22-s + 3.61e3·23-s + 3.12e3·25-s + 460·26-s + 1.12e3·29-s − 5.22e3·31-s + 6.88e3·34-s − 1.55e3·35-s + 1.98e4·37-s + 1.48e4·38-s − 1.34e3·40-s + 1.07e4·41-s + 1.97e4·43-s + 1.44e4·46-s + 9.98e3·47-s + 1.68e4·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.375·5-s − 0.570·7-s − 0.353·8-s + 0.265·10-s + 0.672·11-s + 0.188·13-s − 0.403·14-s − 1/4·16-s + 1.44·17-s + 2.35·19-s + 0.475·22-s + 1.42·23-s + 25-s + 0.133·26-s + 0.248·29-s − 0.977·31-s + 1.02·34-s − 0.214·35-s + 2.38·37-s + 1.66·38-s − 0.132·40-s + 0.999·41-s + 1.62·43-s + 1.00·46-s + 0.659·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.865631211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.865631211\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 21 T - 2684 T^{2} - 21 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 74 T - 11331 T^{2} + 74 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 270 T - 88151 T^{2} - 270 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 115 T - 358068 T^{2} - 115 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 861 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 1850 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3618 T + 6653581 T^{2} - 3618 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1125 T - 19245524 T^{2} - 1125 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5228 T - 1297167 T^{2} + 5228 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 9917 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10758 T - 121637 T^{2} - 10758 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 19714 T + 241633353 T^{2} - 19714 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9984 T - 129664751 T^{2} - 9984 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 36726 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 26460 T - 14792699 T^{2} - 26460 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 53779 T + 2047584540 T^{2} - 53779 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12934 T - 1182836751 T^{2} - 12934 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4254 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 17521 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 36946 T - 1712049483 T^{2} - 36946 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 76416 T + 1900364413 T^{2} + 76416 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 45357 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 127574 T + 7687785219 T^{2} + 127574 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.43677520672843157191355478229, −11.77649338160192739964392328820, −11.12789728413563748121683087718, −11.06499299312297344241373954667, −9.875665508979727914687583626304, −9.823731236484416085588895077682, −9.112349428481126860369183301729, −8.976526912510018265535300869185, −7.67750167086320126910260620381, −7.66595378096261566578945713784, −6.81730851334230360094365819710, −6.23584730423760330171854420872, −5.55280083318196154458011535989, −5.31021804029106814243285728068, −4.46506501583886377640922710062, −3.70614651087798653355550999936, −3.10064427571100045277154312346, −2.63896146940599842871309395604, −1.02254176578052086964751214047, −0.999654497416168277781364818745,
0.999654497416168277781364818745, 1.02254176578052086964751214047, 2.63896146940599842871309395604, 3.10064427571100045277154312346, 3.70614651087798653355550999936, 4.46506501583886377640922710062, 5.31021804029106814243285728068, 5.55280083318196154458011535989, 6.23584730423760330171854420872, 6.81730851334230360094365819710, 7.66595378096261566578945713784, 7.67750167086320126910260620381, 8.976526912510018265535300869185, 9.112349428481126860369183301729, 9.823731236484416085588895077682, 9.875665508979727914687583626304, 11.06499299312297344241373954667, 11.12789728413563748121683087718, 11.77649338160192739964392328820, 12.43677520672843157191355478229
