L(s) = 1 | − 3·9-s + 14·13-s − 20·37-s − 13·49-s − 2·61-s + 20·73-s + 9·81-s + 38·97-s − 34·109-s − 42·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | − 9-s + 3.88·13-s − 3.28·37-s − 1.85·49-s − 0.256·61-s + 2.34·73-s + 81-s + 3.85·97-s − 3.25·109-s − 3.88·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.201020340\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.201020340\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2} \) |
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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
−10.05769130654097618758111806067, −9.270005683140784836591780933170, −9.069612726620187540717790281186, −8.701699375239207520995523548335, −8.402956796952966865129499966223, −8.054530776839527414236421257152, −7.70766500544211890785719330700, −6.80832493091351651812563599795, −6.47864904822841063372522390551, −6.34559182124706036108067872100, −5.80713185390175186171157190897, −5.19039680580781884217894083783, −5.15650859983305851262442063608, −3.99335382965830650528329180599, −3.87370064190643972232271268996, −3.22351797994124113868420109577, −3.12101046085674015410215286326, −1.87657160997623496945792874564, −1.57078476498766989933462670894, −0.66806757366720855191734551195,
0.66806757366720855191734551195, 1.57078476498766989933462670894, 1.87657160997623496945792874564, 3.12101046085674015410215286326, 3.22351797994124113868420109577, 3.87370064190643972232271268996, 3.99335382965830650528329180599, 5.15650859983305851262442063608, 5.19039680580781884217894083783, 5.80713185390175186171157190897, 6.34559182124706036108067872100, 6.47864904822841063372522390551, 6.80832493091351651812563599795, 7.70766500544211890785719330700, 8.054530776839527414236421257152, 8.402956796952966865129499966223, 8.701699375239207520995523548335, 9.069612726620187540717790281186, 9.270005683140784836591780933170, 10.05769130654097618758111806067