| L(s) = 1 | + 15·4-s + 94·11-s + 161·16-s + 112·19-s − 314·29-s + 450·31-s − 280·41-s + 1.41e3·44-s + 650·49-s + 1.49e3·59-s − 676·61-s + 1.45e3·64-s − 64·71-s + 1.68e3·76-s + 2.51e3·79-s − 2.97e3·89-s + 2.67e3·101-s + 1.40e3·109-s − 4.71e3·116-s + 3.96e3·121-s + 6.75e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 15/8·4-s + 2.57·11-s + 2.51·16-s + 1.35·19-s − 2.01·29-s + 2.60·31-s − 1.06·41-s + 4.83·44-s + 1.89·49-s + 3.30·59-s − 1.41·61-s + 2.84·64-s − 0.106·71-s + 2.53·76-s + 3.58·79-s − 3.54·89-s + 2.63·101-s + 1.23·109-s − 3.76·116-s + 2.97·121-s + 4.88·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.579715689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.579715689\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 47 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4369 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7335 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24325 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 157 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 225 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 96406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1405 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 87237 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297738 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 748 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 338 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 359462 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 162866 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1257 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1133170 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1488 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 876670 T^{2} + p^{6} T^{4} \) |
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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
−10.22641667513026760661614720668, −10.03284943228595873449160711991, −9.326066294285412910104868816773, −9.308148079152604263332168623621, −8.351501922798553568534662672516, −8.288658306347035483104371569324, −7.31663178586811185764588145020, −7.28893353346088746275436031817, −6.73863888401773663380906319697, −6.50349056041767526985503177566, −5.79397257450923577744621128540, −5.74251631324208817724629205805, −4.84182730047314026617280511959, −4.13736897755266664536595431258, −3.45363015392713066837879804081, −3.41290813098502162031675107881, −2.38872405900784799625724881480, −2.01573038662052173257969580933, −1.08122803189843617686438808898, −1.03899341066179496681632477488,
1.03899341066179496681632477488, 1.08122803189843617686438808898, 2.01573038662052173257969580933, 2.38872405900784799625724881480, 3.41290813098502162031675107881, 3.45363015392713066837879804081, 4.13736897755266664536595431258, 4.84182730047314026617280511959, 5.74251631324208817724629205805, 5.79397257450923577744621128540, 6.50349056041767526985503177566, 6.73863888401773663380906319697, 7.28893353346088746275436031817, 7.31663178586811185764588145020, 8.288658306347035483104371569324, 8.351501922798553568534662672516, 9.308148079152604263332168623621, 9.326066294285412910104868816773, 10.03284943228595873449160711991, 10.22641667513026760661614720668
