| L(s) = 1 | − 16·4-s + 192·16-s − 250·25-s − 616·31-s − 220·37-s + 286·49-s − 2.04e3·64-s + 1.76e3·67-s − 2.66e3·97-s + 4.00e3·100-s + 3.64e3·103-s + 9.85e3·124-s + 127-s + 131-s + 137-s + 139-s + 3.52e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 2·25-s − 3.56·31-s − 0.977·37-s + 0.833·49-s − 4·64-s + 3.20·67-s − 2.78·97-s + 4·100-s + 3.48·103-s + 7.13·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 1.95·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2811803650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2811803650\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10582 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 308 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 110 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 111386 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 420838 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 638066 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 204622 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1330 T + p^{3} T^{2} )^{2} \) |
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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
−9.503810797076129381759282651921, −9.344106994820856639948554638967, −8.900471737436137317607565865500, −8.628538559741851990340246354690, −7.974761661277483626431626114359, −7.88785451652796814560835957300, −7.26753499824721336835387423903, −6.91969628393304027803761574933, −6.10723653097638961641167551700, −5.64943836005192518837278468000, −5.31533651828802169768313839251, −5.16927463903276442719342342457, −4.22340460746868986332528628504, −4.12790738348183243676044079602, −3.48615127189063176105368371979, −3.37783782367391371379408606514, −2.14745526970691356102549916476, −1.75983754599736757365364074390, −0.873374157532242061903856835480, −0.17209577057262730509541161387,
0.17209577057262730509541161387, 0.873374157532242061903856835480, 1.75983754599736757365364074390, 2.14745526970691356102549916476, 3.37783782367391371379408606514, 3.48615127189063176105368371979, 4.12790738348183243676044079602, 4.22340460746868986332528628504, 5.16927463903276442719342342457, 5.31533651828802169768313839251, 5.64943836005192518837278468000, 6.10723653097638961641167551700, 6.91969628393304027803761574933, 7.26753499824721336835387423903, 7.88785451652796814560835957300, 7.974761661277483626431626114359, 8.628538559741851990340246354690, 8.900471737436137317607565865500, 9.344106994820856639948554638967, 9.503810797076129381759282651921
