| L(s) = 1 | − 1.22e3·5-s + 3.13e3·9-s + 1.09e4·13-s + 1.46e5·17-s + 3.35e5·25-s + 2.56e5·29-s + 6.94e6·37-s + 4.29e6·41-s − 3.82e6·45-s + 9.57e6·49-s − 1.64e6·53-s + 2.94e7·61-s − 1.33e7·65-s − 1.14e7·73-s − 3.31e7·81-s − 1.78e8·85-s − 1.66e8·89-s + 2.41e8·97-s − 5.54e7·101-s + 1.18e8·109-s + 1.10e8·113-s + 3.43e7·117-s + 8.70e7·121-s + 5.21e8·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.95·5-s + 0.478·9-s + 0.383·13-s + 1.75·17-s + 0.857·25-s + 0.362·29-s + 3.70·37-s + 1.51·41-s − 0.933·45-s + 1.66·49-s − 0.208·53-s + 2.13·61-s − 0.747·65-s − 0.403·73-s − 0.771·81-s − 3.41·85-s − 2.65·89-s + 2.72·97-s − 0.532·101-s + 0.836·109-s + 0.675·113-s + 0.183·117-s + 0.405·121-s + 2.13·125-s − 0.707·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.176971988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.176971988\) |
| \(L(5)\) |
|
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 | $C_2^2$ | \( 1 - 1046 p T^{2} + p^{16} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 122 p T + p^{8} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 195362 p^{2} T^{2} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 87015362 T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5470 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 73090 T + p^{8} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 33587484482 T^{2} + p^{16} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 100353384578 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 128222 T + p^{8} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1701165473282 T^{2} + p^{16} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3472030 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2146882 T + p^{8} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 11766582970942 T^{2} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10537750788862 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 824290 T + p^{8} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 279781405698242 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14746078 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 579369070794818 T^{2} + p^{16} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1290076545985922 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 5725630 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1744457179595522 T^{2} + p^{16} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1804713576833858 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 83324222 T + p^{8} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 120619010 T + p^{8} T^{2} )^{2} \) |
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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
−13.37616462565233790914174864650, −12.80762194927807919640080796236, −12.35772315185534093356642805183, −11.75958510373183490993354768687, −11.37172840882685673283418194405, −10.92897962510779414294763752497, −9.923664093719332757652865314180, −9.702490276936449846144991313921, −8.633702945814849767089338009083, −8.101762591965476280994931162787, −7.44751286963590300316291945565, −7.43448164004723637703366343760, −6.17231524327664394917947694639, −5.59513837838510094416668767674, −4.37337893284173940252729839676, −4.12442295323724367270428988216, −3.38750645087186340246926491883, −2.50993112200588389860417846615, −1.03321725869978853116927577304, −0.63307275723312919305251802875,
0.63307275723312919305251802875, 1.03321725869978853116927577304, 2.50993112200588389860417846615, 3.38750645087186340246926491883, 4.12442295323724367270428988216, 4.37337893284173940252729839676, 5.59513837838510094416668767674, 6.17231524327664394917947694639, 7.43448164004723637703366343760, 7.44751286963590300316291945565, 8.101762591965476280994931162787, 8.633702945814849767089338009083, 9.702490276936449846144991313921, 9.923664093719332757652865314180, 10.92897962510779414294763752497, 11.37172840882685673283418194405, 11.75958510373183490993354768687, 12.35772315185534093356642805183, 12.80762194927807919640080796236, 13.37616462565233790914174864650
