| L(s) = 1 | − 2·4-s − 10·7-s + 3·16-s − 14·25-s + 20·28-s − 8·37-s − 8·43-s + 61·49-s − 4·64-s + 8·67-s + 32·79-s + 28·100-s − 56·109-s − 30·112-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 44·169-s + 16·172-s + ⋯ |
| L(s) = 1 | − 4-s − 3.77·7-s + 3/4·16-s − 2.79·25-s + 3.77·28-s − 1.31·37-s − 1.21·43-s + 61/7·49-s − 1/2·64-s + 0.977·67-s + 3.60·79-s + 14/5·100-s − 5.36·109-s − 2.83·112-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.38·169-s + 1.21·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1091472496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1091472496\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 163 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.213747117765628748695473348112, −8.142899509738087537620191589828, −7.77949847293907118173619831348, −7.44445268402087159160073099768, −7.16107612388200978021210564818, −6.85315420669785975576188112458, −6.70024093427145763703808662451, −6.59170704931012499048092969127, −6.10793565913202028806178696743, −5.91496784936962241399101036041, −5.91046636029868485221153810852, −5.42256276793357544650064973651, −5.19386860282878493888479508112, −4.96325070707529899734466334736, −4.29456560128739808844523785185, −4.17678286098748967204483568362, −3.68526917680665035907942788732, −3.60271748650094200697432923611, −3.55858479146466087117254049237, −2.93287923154443980205989276092, −2.86694919426937825445757505001, −2.12288595849623451415052560578, −1.97329068957981973631281959760, −0.899611126626033953425390149797, −0.16688697559101142368562604544,
0.16688697559101142368562604544, 0.899611126626033953425390149797, 1.97329068957981973631281959760, 2.12288595849623451415052560578, 2.86694919426937825445757505001, 2.93287923154443980205989276092, 3.55858479146466087117254049237, 3.60271748650094200697432923611, 3.68526917680665035907942788732, 4.17678286098748967204483568362, 4.29456560128739808844523785185, 4.96325070707529899734466334736, 5.19386860282878493888479508112, 5.42256276793357544650064973651, 5.91046636029868485221153810852, 5.91496784936962241399101036041, 6.10793565913202028806178696743, 6.59170704931012499048092969127, 6.70024093427145763703808662451, 6.85315420669785975576188112458, 7.16107612388200978021210564818, 7.44445268402087159160073099768, 7.77949847293907118173619831348, 8.142899509738087537620191589828, 8.213747117765628748695473348112
