L(s) = 1 | + 2·3-s − 3·4-s − 4·5-s + 3·9-s − 6·12-s − 8·15-s + 5·16-s + 12·20-s + 8·23-s + 2·25-s + 4·27-s + 16·31-s − 9·36-s − 20·37-s − 12·45-s + 24·47-s + 10·48-s − 14·49-s − 12·53-s − 24·59-s + 24·60-s − 3·64-s − 8·67-s + 16·69-s + 4·75-s − 20·80-s + 5·81-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 3/2·4-s − 1.78·5-s + 9-s − 1.73·12-s − 2.06·15-s + 5/4·16-s + 2.68·20-s + 1.66·23-s + 2/5·25-s + 0.769·27-s + 2.87·31-s − 3/2·36-s − 3.28·37-s − 1.78·45-s + 3.50·47-s + 1.44·48-s − 2·49-s − 1.64·53-s − 3.12·59-s + 3.09·60-s − 3/8·64-s − 0.977·67-s + 1.92·69-s + 0.461·75-s − 2.23·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9843407694\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9843407694\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.653029479594481171374089128382, −8.263513432862467471567669978548, −7.86695452924192516668672252135, −7.52311475462874440391759117602, −7.09492951839214399611242033655, −6.46787589567348650226556085189, −5.77601443425071364494220587649, −4.88480893363690614197292172734, −4.56991321080597356703354311172, −4.43128249952073642221709631783, −3.48982533128044565942915605098, −3.44365518362789223021999993571, −2.81805056436224670768266257292, −1.60994939898295228077229523210, −0.54470855901702999376946866933,
0.54470855901702999376946866933, 1.60994939898295228077229523210, 2.81805056436224670768266257292, 3.44365518362789223021999993571, 3.48982533128044565942915605098, 4.43128249952073642221709631783, 4.56991321080597356703354311172, 4.88480893363690614197292172734, 5.77601443425071364494220587649, 6.46787589567348650226556085189, 7.09492951839214399611242033655, 7.52311475462874440391759117602, 7.86695452924192516668672252135, 8.263513432862467471567669978548, 8.653029479594481171374089128382