| L(s) = 1 | + 4·5-s − 4·9-s + 12·13-s + 2·25-s + 4·29-s + 12·37-s + 12·41-s − 16·45-s − 6·49-s − 4·53-s + 12·61-s + 48·65-s − 24·73-s + 7·81-s − 24·89-s − 16·97-s − 20·101-s − 12·109-s + 12·113-s − 48·117-s − 4·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 4/3·9-s + 3.32·13-s + 2/5·25-s + 0.742·29-s + 1.97·37-s + 1.87·41-s − 2.38·45-s − 6/7·49-s − 0.549·53-s + 1.53·61-s + 5.95·65-s − 2.80·73-s + 7/9·81-s − 2.54·89-s − 1.62·97-s − 1.99·101-s − 1.14·109-s + 1.12·113-s − 4.43·117-s − 0.363·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.720951236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.720951236\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−11.00418174970039750289559385072, −10.88820804870763015630312509868, −10.04203083922318676013244980209, −9.939496571832144260695678868519, −9.171106386262901820522468309528, −9.036800116186650427672448385071, −8.504509402954959248433237085570, −8.136209404913370480790908559261, −7.71500601574158399637037578135, −6.61387298753804977066819083208, −6.39786281520133230964526164360, −5.98859826938756647506654502941, −5.59569989859156343397361984183, −5.47761574613046253906233273467, −4.17878088743430670818200841248, −4.03141419798265746295869587755, −2.93642251168937868637046890158, −2.73263639610669961746742872748, −1.68277690453840397387089625321, −1.12736751813594213332646162785,
1.12736751813594213332646162785, 1.68277690453840397387089625321, 2.73263639610669961746742872748, 2.93642251168937868637046890158, 4.03141419798265746295869587755, 4.17878088743430670818200841248, 5.47761574613046253906233273467, 5.59569989859156343397361984183, 5.98859826938756647506654502941, 6.39786281520133230964526164360, 6.61387298753804977066819083208, 7.71500601574158399637037578135, 8.136209404913370480790908559261, 8.504509402954959248433237085570, 9.036800116186650427672448385071, 9.171106386262901820522468309528, 9.939496571832144260695678868519, 10.04203083922318676013244980209, 10.88820804870763015630312509868, 11.00418174970039750289559385072
