| L(s) = 1 | − 2·4-s − 2·7-s − 2·9-s + 6·11-s − 16·13-s + 3·16-s + 4·17-s − 20·19-s − 2·25-s + 4·28-s + 4·36-s + 16·37-s − 40·41-s − 12·44-s + 2·49-s + 32·52-s − 4·53-s − 4·61-s + 4·63-s − 4·64-s − 32·67-s − 8·68-s − 8·71-s − 20·73-s + 40·76-s − 12·77-s + 3·81-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s − 2/3·9-s + 1.80·11-s − 4.43·13-s + 3/4·16-s + 0.970·17-s − 4.58·19-s − 2/5·25-s + 0.755·28-s + 2/3·36-s + 2.63·37-s − 6.24·41-s − 1.80·44-s + 2/7·49-s + 4.43·52-s − 0.549·53-s − 0.512·61-s + 0.503·63-s − 1/2·64-s − 3.90·67-s − 0.970·68-s − 0.949·71-s − 2.34·73-s + 4.58·76-s − 1.36·77-s + 1/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3566 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 10766 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 110 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 17254 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 168 T^{2} + 18686 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 11590 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.92955226847249353743298325884, −6.38504165409295321102293484439, −6.33938469078850582285032608185, −6.27326049899335608978690121381, −6.23908084266381531966183464683, −5.67359786997772698281840866090, −5.59920388913470797281007512257, −5.31208020334116060789815084907, −4.99791390307651633650140741724, −4.74103279367040997635844961344, −4.73947273516567680643853648119, −4.58639361000392654772655749363, −4.36912775291010658399612083172, −4.10863810879050391768937483961, −3.75596920492615302953271537259, −3.63046455285934971636457736372, −3.54995852589984067401823986965, −2.85201948065146787987541052879, −2.77655652166459801319096682006, −2.64490646406001688716169274109, −2.53004061862695530445585287804, −1.95595002387527657358053509960, −1.67687621235150514883212676605, −1.63174736288633383289351908059, −1.14525282390441435193514863756, 0, 0, 0, 0,
1.14525282390441435193514863756, 1.63174736288633383289351908059, 1.67687621235150514883212676605, 1.95595002387527657358053509960, 2.53004061862695530445585287804, 2.64490646406001688716169274109, 2.77655652166459801319096682006, 2.85201948065146787987541052879, 3.54995852589984067401823986965, 3.63046455285934971636457736372, 3.75596920492615302953271537259, 4.10863810879050391768937483961, 4.36912775291010658399612083172, 4.58639361000392654772655749363, 4.73947273516567680643853648119, 4.74103279367040997635844961344, 4.99791390307651633650140741724, 5.31208020334116060789815084907, 5.59920388913470797281007512257, 5.67359786997772698281840866090, 6.23908084266381531966183464683, 6.27326049899335608978690121381, 6.33938469078850582285032608185, 6.38504165409295321102293484439, 6.92955226847249353743298325884
