| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5·5-s − 2·6-s + 3·7-s − 5·8-s + 10·10-s − 11·11-s + 2·12-s − 6·13-s − 6·14-s − 5·15-s + 5·16-s − 2·17-s − 10·20-s + 3·21-s + 22·22-s − 6·23-s − 5·24-s + 10·25-s + 12·26-s + 6·28-s − 15·29-s + 10·30-s − 32·31-s + 2·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 2.23·5-s − 0.816·6-s + 1.13·7-s − 1.76·8-s + 3.16·10-s − 3.31·11-s + 0.577·12-s − 1.66·13-s − 1.60·14-s − 1.29·15-s + 5/4·16-s − 0.485·17-s − 2.23·20-s + 0.654·21-s + 4.69·22-s − 1.25·23-s − 1.02·24-s + 2·25-s + 2.35·26-s + 1.13·28-s − 2.78·29-s + 1.82·30-s − 5.74·31-s + 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + p T + 51 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 6 T + 3 T^{2} - 10 T^{3} + 81 T^{4} - 10 p T^{5} + 3 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 2 T + 7 T^{2} + 70 T^{3} + 441 T^{4} + 70 p T^{5} + 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 9 T^{2} + 70 T^{3} + 291 T^{4} + 70 p T^{5} - 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 6 T + 13 T^{2} + 150 T^{3} + 1231 T^{4} + 150 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 15 T + 161 T^{2} + 1245 T^{3} + 7636 T^{4} + 1245 p T^{5} + 161 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 16 T + 121 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 2 T + 27 T^{2} + 160 T^{3} + 1841 T^{4} + 160 p T^{5} + 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 8 T + 67 T^{2} - 610 T^{3} + 6231 T^{4} - 610 p T^{5} + 67 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 4 T + 43 T^{2} - 110 T^{3} + 741 T^{4} - 110 p T^{5} + 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 20 T + 101 T^{2} + 20 p T^{3} - 18439 T^{4} + 20 p^{2} T^{5} + 101 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 17 T + 78 T^{2} + 289 T^{3} + 3755 T^{4} + 289 p T^{5} + 78 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_4\times C_2$ | \( 1 - 3 T - 58 T^{2} + 375 T^{3} + 2761 T^{4} + 375 p T^{5} - 58 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 21 T + 251 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_4\times C_2$ | \( 1 - 5 T - 54 T^{2} + 665 T^{3} + 941 T^{4} + 665 p T^{5} - 54 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 4 T - 37 T^{2} + 710 T^{3} + 851 T^{4} + 710 p T^{5} - 37 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 5 T - 74 T^{2} - 25 T^{3} + 8391 T^{4} - 25 p T^{5} - 74 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 3 T + 12 T^{2} - 865 T^{3} + 11511 T^{4} - 865 p T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.81269040068691720105701465366, −7.63110964569028031480770993099, −7.40413127521042526104119186891, −7.32123161334087670419230565092, −7.31274018726423540538340092732, −6.93685456804716576307987708144, −6.80366401123376073657738098065, −6.08393998742236711450600402396, −5.75650631973711278050269755000, −5.65830807403735957200051084726, −5.58858926336970791521682943407, −5.17097538236043121501927414858, −5.03338986418709786767891618844, −4.94528814010225004259915737141, −4.27836152520680742291100880135, −4.11617667248104771668021516535, −3.80653535789610977788320710313, −3.74921049426773414303443995810, −3.47953928792298006864853723238, −2.89385954104433795474517428164, −2.58860227720906003339463371462, −2.45122206096378338482128187014, −2.31819851287167964098382944437, −1.74930780554351980452551852782, −1.56716687280036555002736777719, 0, 0, 0, 0,
1.56716687280036555002736777719, 1.74930780554351980452551852782, 2.31819851287167964098382944437, 2.45122206096378338482128187014, 2.58860227720906003339463371462, 2.89385954104433795474517428164, 3.47953928792298006864853723238, 3.74921049426773414303443995810, 3.80653535789610977788320710313, 4.11617667248104771668021516535, 4.27836152520680742291100880135, 4.94528814010225004259915737141, 5.03338986418709786767891618844, 5.17097538236043121501927414858, 5.58858926336970791521682943407, 5.65830807403735957200051084726, 5.75650631973711278050269755000, 6.08393998742236711450600402396, 6.80366401123376073657738098065, 6.93685456804716576307987708144, 7.31274018726423540538340092732, 7.32123161334087670419230565092, 7.40413127521042526104119186891, 7.63110964569028031480770993099, 7.81269040068691720105701465366
