| L(s) = 1 | + 960·4-s + 2.63e4·9-s − 9.24e4·11-s + 6.59e5·16-s + 2.01e6·19-s − 8.39e6·29-s − 6.73e6·31-s + 2.53e7·36-s + 2.21e7·41-s − 8.87e7·44-s + 6.27e7·49-s + 1.70e8·59-s + 9.14e7·61-s + 3.81e8·64-s − 3.79e8·71-s + 1.93e9·76-s − 1.90e8·79-s + 3.07e8·81-s + 3.98e7·89-s − 2.43e9·99-s − 4.33e8·101-s + 4.99e9·109-s − 8.05e9·116-s + 1.68e9·121-s − 6.46e9·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 15/8·4-s + 1.33·9-s − 1.90·11-s + 2.51·16-s + 3.55·19-s − 2.20·29-s − 1.30·31-s + 2.51·36-s + 1.22·41-s − 3.56·44-s + 1.55·49-s + 1.83·59-s + 0.846·61-s + 2.84·64-s − 1.77·71-s + 6.65·76-s − 0.549·79-s + 0.794·81-s + 0.0673·89-s − 2.54·99-s − 0.414·101-s + 3.38·109-s − 4.13·116-s + 0.716·121-s − 2.45·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.602553429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.602553429\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 p^{6} T^{2} + p^{18} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2930 p^{2} T^{2} + p^{18} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1279850 p^{2} T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 46208 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 271990 p^{4} T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7692851970 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 1008740 T + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3318691560010 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4196390 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3365028 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36978027865990 T^{2} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 11056262 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 964266250395250 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 973807228536570 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5020371966324870 T^{2} + p^{18} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 85185620 T + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 45748642 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 52362232686188930 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 189967468 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 52141715309999090 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 95040840 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 305390309466662530 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 19938630 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1520081736335854270 T^{2} + p^{18} T^{4} \) |
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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
−15.78543726481236420532012830190, −15.48850309706122149045382192929, −14.81353919536456208711925480577, −13.90394908824848319641188278895, −12.92781963465491192705091219246, −12.79592969474406592692231168612, −11.63363158342774976541999988269, −11.44469568079794094733153013072, −10.51076079669956257472503771532, −10.09157375767329676979363775729, −9.338219174520221660525468675806, −7.71319036573704359358537929929, −7.39235729187329437434913479409, −7.19572065102375042778024019768, −5.58060200542851640227041726615, −5.45056437636152025172754765625, −3.65359882793067861220445820694, −2.79485294626219371182620123805, −1.89821315518625510633248891751, −0.954598983006615405882800477889,
0.954598983006615405882800477889, 1.89821315518625510633248891751, 2.79485294626219371182620123805, 3.65359882793067861220445820694, 5.45056437636152025172754765625, 5.58060200542851640227041726615, 7.19572065102375042778024019768, 7.39235729187329437434913479409, 7.71319036573704359358537929929, 9.338219174520221660525468675806, 10.09157375767329676979363775729, 10.51076079669956257472503771532, 11.44469568079794094733153013072, 11.63363158342774976541999988269, 12.79592969474406592692231168612, 12.92781963465491192705091219246, 13.90394908824848319641188278895, 14.81353919536456208711925480577, 15.48850309706122149045382192929, 15.78543726481236420532012830190
