L(s) = 1 | + 2·3-s + 5-s + 3·9-s + 12·13-s + 2·15-s + 25-s + 4·27-s − 16·31-s − 4·37-s + 24·39-s − 12·41-s + 24·43-s + 3·45-s − 14·49-s + 12·53-s + 12·65-s + 8·67-s + 16·71-s + 2·75-s − 16·79-s + 5·81-s − 24·83-s + 20·89-s − 32·93-s − 8·107-s − 8·111-s + 36·117-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 9-s + 3.32·13-s + 0.516·15-s + 1/5·25-s + 0.769·27-s − 2.87·31-s − 0.657·37-s + 3.84·39-s − 1.87·41-s + 3.65·43-s + 0.447·45-s − 2·49-s + 1.64·53-s + 1.48·65-s + 0.977·67-s + 1.89·71-s + 0.230·75-s − 1.80·79-s + 5/9·81-s − 2.63·83-s + 2.11·89-s − 3.31·93-s − 0.773·107-s − 0.759·111-s + 3.32·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.077101115\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.077101115\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.323057177226078640880314002796, −8.272894657123258864282541087163, −7.66606125131759218514172865215, −6.89270810769313104844169224865, −6.88205984732646038789481985379, −6.10521792180854763107561075146, −5.59585158632682227968586342616, −5.47564398271108061480499478420, −4.38946567062600160988680606034, −3.95303834850677407956821293662, −3.48838247700931708875876876915, −3.25163337009512069140834562852, −2.25002432004047770108316173263, −1.70048286389009539990984644895, −1.09686843848459544952208950048,
1.09686843848459544952208950048, 1.70048286389009539990984644895, 2.25002432004047770108316173263, 3.25163337009512069140834562852, 3.48838247700931708875876876915, 3.95303834850677407956821293662, 4.38946567062600160988680606034, 5.47564398271108061480499478420, 5.59585158632682227968586342616, 6.10521792180854763107561075146, 6.88205984732646038789481985379, 6.89270810769313104844169224865, 7.66606125131759218514172865215, 8.272894657123258864282541087163, 8.323057177226078640880314002796