| L(s) = 1 | − 12·5-s − 243·9-s + 5.30e3·13-s − 1.44e4·17-s − 3.11e4·25-s − 2.31e4·29-s − 4.46e4·37-s + 2.07e5·41-s + 2.91e3·45-s + 1.94e5·49-s − 3.36e5·53-s + 5.20e5·61-s − 6.36e4·65-s − 7.91e5·73-s + 5.90e4·81-s + 1.72e5·85-s − 5.03e5·89-s + 1.03e6·97-s − 2.04e6·101-s − 1.90e6·109-s − 2.42e5·113-s − 1.28e6·117-s + 3.12e6·121-s + 5.61e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.0959·5-s − 1/3·9-s + 2.41·13-s − 2.93·17-s − 1.99·25-s − 0.947·29-s − 0.882·37-s + 3.00·41-s + 0.0319·45-s + 1.65·49-s − 2.26·53-s + 2.29·61-s − 0.231·65-s − 2.03·73-s + 1/9·81-s + 0.281·85-s − 0.714·89-s + 1.13·97-s − 1.98·101-s − 1.47·109-s − 0.167·113-s − 0.805·117-s + 1.76·121-s + 0.287·125-s + 0.0909·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.125293157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125293157\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 6 T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 194930 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3127970 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2654 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7206 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 81557954 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8747422 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 11550 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1702033490 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 22346 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 103626 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3585198814 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4308279550 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 168462 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 20657330 p^{2} T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 70 p^{2} T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 80375086466 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 245662064350 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 395918 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 176211604754 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 645933881666 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 251886 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 517474 T + p^{6} T^{2} )^{2} \) |
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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
−11.37154169502675660076633119485, −11.22037719508825778687115133056, −10.94095689565675097780972874818, −10.41045307311605682857781797228, −9.479291894298240860542527200316, −9.184284158197369843124791145034, −8.672389780173542911616842460442, −8.328406724940101714165745960299, −7.67105656188679982942401945587, −7.01706311405932109915439544591, −6.41246342334536168031765838371, −5.94012018158778903955430143532, −5.61847692480391319823178238113, −4.50258364674187048120288851987, −4.01522902089987152663476470717, −3.70583935074325361871733953681, −2.60575589446991899657678372333, −2.02802944509387406725098201226, −1.28407803668833243997800856541, −0.29885969161834938913028930533,
0.29885969161834938913028930533, 1.28407803668833243997800856541, 2.02802944509387406725098201226, 2.60575589446991899657678372333, 3.70583935074325361871733953681, 4.01522902089987152663476470717, 4.50258364674187048120288851987, 5.61847692480391319823178238113, 5.94012018158778903955430143532, 6.41246342334536168031765838371, 7.01706311405932109915439544591, 7.67105656188679982942401945587, 8.328406724940101714165745960299, 8.672389780173542911616842460442, 9.184284158197369843124791145034, 9.479291894298240860542527200316, 10.41045307311605682857781797228, 10.94095689565675097780972874818, 11.22037719508825778687115133056, 11.37154169502675660076633119485
