| L(s) = 1 | − 4·3-s + 3·4-s + 6·9-s − 12·12-s + 4·13-s + 5·16-s − 12·23-s + 4·27-s − 12·29-s + 18·36-s − 16·39-s + 12·43-s − 20·48-s + 14·49-s + 12·52-s + 24·53-s + 12·61-s + 3·64-s + 48·69-s − 37·81-s + 48·87-s − 36·92-s + 12·101-s − 12·103-s − 12·107-s + 12·108-s − 36·116-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3/2·4-s + 2·9-s − 3.46·12-s + 1.10·13-s + 5/4·16-s − 2.50·23-s + 0.769·27-s − 2.22·29-s + 3·36-s − 2.56·39-s + 1.82·43-s − 2.88·48-s + 2·49-s + 1.66·52-s + 3.29·53-s + 1.53·61-s + 3/8·64-s + 5.77·69-s − 4.11·81-s + 5.14·87-s − 3.75·92-s + 1.19·101-s − 1.18·103-s − 1.16·107-s + 1.15·108-s − 3.34·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8576723380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8576723380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−11.99296353276332942642765155921, −11.42992459237653641999388807895, −11.01254498328097680870653599007, −10.65534114512796297854884259577, −10.37743657056284865208759440962, −9.882562489060455031237515278807, −9.022312494215980019091934159935, −8.531257305818010614689127322682, −7.83680075981377800263635606552, −7.25335192544233749893786306720, −6.93064912686883245100387620651, −6.19833588257915813134827660667, −6.02407049889489475068721471980, −5.51158480258020253772074522027, −5.44260138555209788819174274601, −4.01076134625495095606087851020, −3.96864114407920061231982211201, −2.60985512727022564354621993941, −1.93515407303545731145842992812, −0.75437596206238912134373358332,
0.75437596206238912134373358332, 1.93515407303545731145842992812, 2.60985512727022564354621993941, 3.96864114407920061231982211201, 4.01076134625495095606087851020, 5.44260138555209788819174274601, 5.51158480258020253772074522027, 6.02407049889489475068721471980, 6.19833588257915813134827660667, 6.93064912686883245100387620651, 7.25335192544233749893786306720, 7.83680075981377800263635606552, 8.531257305818010614689127322682, 9.022312494215980019091934159935, 9.882562489060455031237515278807, 10.37743657056284865208759440962, 10.65534114512796297854884259577, 11.01254498328097680870653599007, 11.42992459237653641999388807895, 11.99296353276332942642765155921
