| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s − 8·5-s + 16·6-s − 16·8-s + 8·9-s + 32·10-s − 32·12-s + 32·15-s + 36·16-s − 16·17-s − 32·18-s − 16·19-s − 64·20-s − 4·23-s + 64·24-s + 32·25-s − 12·27-s − 128·30-s − 8·31-s − 64·32-s + 64·34-s + 64·36-s − 20·37-s + 64·38-s + 128·40-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s − 3.57·5-s + 6.53·6-s − 5.65·8-s + 8/3·9-s + 10.1·10-s − 9.23·12-s + 8.26·15-s + 9·16-s − 3.88·17-s − 7.54·18-s − 3.67·19-s − 14.3·20-s − 0.834·23-s + 13.0·24-s + 32/5·25-s − 2.30·27-s − 23.3·30-s − 1.43·31-s − 11.3·32-s + 10.9·34-s + 32/3·36-s − 3.28·37-s + 10.3·38-s + 20.2·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 96 T^{3} + 239 T^{4} + 96 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 672 T^{3} + 2999 T^{4} + 672 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 2179 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 80 T^{3} - 1481 T^{4} - 80 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 11662 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 592 T^{3} + 2722 T^{4} - 592 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 6379 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1120 T^{3} + 9271 T^{4} - 1120 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 46 T^{2} + 5635 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 6034 T^{4} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 200 T^{3} - 7214 T^{4} - 200 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 28 T + 392 T^{2} + 3724 T^{3} + 31534 T^{4} + 3724 p T^{5} + 392 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 576 T^{3} + 10367 T^{4} + 576 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 6 T + 159 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 13447 T^{4} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 6656 T^{3} + 72367 T^{4} + 6656 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 7168 T^{3} + 84223 T^{4} + 7168 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.076734636261340309633416507153, −8.834179103879729929758875331683, −8.581873106039718668217667362342, −8.492484730248469175438484742926, −8.483348024469992215282166338049, −8.097983645952740801896972106241, −7.58809239306743547505090731178, −7.30831629764869960913996894865, −7.27051829409949760496563209021, −7.06615897152807525921747390704, −6.80272288362311982567881558040, −6.22995800568119484029125961470, −6.06268353447512947124190755758, −6.05099925858386684562592065532, −5.93105854588474171452825919770, −5.05720555293111183915009797970, −4.78967987943054034412739943459, −4.43397738359222904070936718564, −4.10688183925757030996443034502, −3.92180989062329451106168951247, −3.86694938889073977520339782817, −3.10758422877358924342168508511, −2.44398454524324804949917424868, −2.27365278627532826967251926893, −1.36491500348493949959368311463, 0, 0, 0, 0,
1.36491500348493949959368311463, 2.27365278627532826967251926893, 2.44398454524324804949917424868, 3.10758422877358924342168508511, 3.86694938889073977520339782817, 3.92180989062329451106168951247, 4.10688183925757030996443034502, 4.43397738359222904070936718564, 4.78967987943054034412739943459, 5.05720555293111183915009797970, 5.93105854588474171452825919770, 6.05099925858386684562592065532, 6.06268353447512947124190755758, 6.22995800568119484029125961470, 6.80272288362311982567881558040, 7.06615897152807525921747390704, 7.27051829409949760496563209021, 7.30831629764869960913996894865, 7.58809239306743547505090731178, 8.097983645952740801896972106241, 8.483348024469992215282166338049, 8.492484730248469175438484742926, 8.581873106039718668217667362342, 8.834179103879729929758875331683, 9.076734636261340309633416507153
