| L(s) = 1 | + 12·9-s + 8·41-s + 12·49-s + 90·81-s + 56·89-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | + 4·9-s + 1.24·41-s + 12/7·49-s + 10·81-s + 5.93·89-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.678742824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.678742824\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.78251091590754585060568078516, −6.36858298557821279831329280348, −6.33569840776684366693071063117, −6.26262096637776380756624938629, −5.98144040077376839836244720847, −5.57962107921070869285111769285, −5.22851186037834574817226728988, −5.16233363972040997814095375736, −4.99470588143299034076198925000, −4.51258153787500963958187834931, −4.44073529687710282253373585693, −4.35658374271407549268342316965, −4.13315366744863959186005773578, −3.88357754307694610488811170521, −3.50486703686459883406727326561, −3.34764965365937316480056464511, −3.28333376198976661924085206796, −2.63193185937014753812212652388, −2.27036148784765714704029151201, −2.03559908497542300023086080287, −1.98277825287775790997044516532, −1.57784132869182798225866603749, −1.06063922479527703628902030604, −0.930536557049461890747862097621, −0.63939751647073664019684694016,
0.63939751647073664019684694016, 0.930536557049461890747862097621, 1.06063922479527703628902030604, 1.57784132869182798225866603749, 1.98277825287775790997044516532, 2.03559908497542300023086080287, 2.27036148784765714704029151201, 2.63193185937014753812212652388, 3.28333376198976661924085206796, 3.34764965365937316480056464511, 3.50486703686459883406727326561, 3.88357754307694610488811170521, 4.13315366744863959186005773578, 4.35658374271407549268342316965, 4.44073529687710282253373585693, 4.51258153787500963958187834931, 4.99470588143299034076198925000, 5.16233363972040997814095375736, 5.22851186037834574817226728988, 5.57962107921070869285111769285, 5.98144040077376839836244720847, 6.26262096637776380756624938629, 6.33569840776684366693071063117, 6.36858298557821279831329280348, 6.78251091590754585060568078516
