| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 7-s + 5·8-s − 2·10-s − 11-s − 2·12-s + 13-s − 2·14-s + 15-s + 5·16-s + 7·17-s − 2·20-s + 21-s − 2·22-s − 4·23-s − 5·24-s + 2·26-s − 2·28-s + 15·29-s + 2·30-s + 13·31-s − 2·32-s + 33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 1.76·8-s − 0.632·10-s − 0.301·11-s − 0.577·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 5/4·16-s + 1.69·17-s − 0.447·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s − 1.02·24-s + 0.392·26-s − 0.377·28-s + 2.78·29-s + 0.365·30-s + 2.33·31-s − 0.353·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.273061281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.273061281\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 11 p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 - T - 12 T^{2} + 25 T^{3} + 131 T^{4} + 25 p T^{5} - 12 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 7 T + 2 T^{2} + 65 T^{3} - 169 T^{4} + 65 p T^{5} + 2 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 15 T + 106 T^{2} - 675 T^{3} + 4171 T^{4} - 675 p T^{5} + 106 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 13 T + 48 T^{2} + 319 T^{3} - 3835 T^{4} + 319 p T^{5} + 48 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 22 T + 207 T^{2} - 1220 T^{3} + 6701 T^{4} - 1220 p T^{5} + 207 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 13 T + 53 T^{2} - 331 T^{3} + 3380 T^{4} - 331 p T^{5} + 53 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 18 T + 137 T^{2} + 990 T^{3} + 7951 T^{4} + 990 p T^{5} + 137 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 9 T - 7 T^{2} - 525 T^{3} - 3884 T^{4} - 525 p T^{5} - 7 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 15 T + 31 T^{2} + 15 p T^{3} - 10424 T^{4} + 15 p^{2} T^{5} + 31 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 3 T + 18 T^{2} + 209 T^{3} + 675 T^{4} + 209 p T^{5} + 18 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 13 T - 2 T^{2} + 349 T^{3} + 405 T^{4} + 349 p T^{5} - 2 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 2595 p T^{5} + 233 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 15 T + 11 T^{2} - 15 p T^{3} - 13064 T^{4} - 15 p^{2} T^{5} + 11 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 780 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 15 T + 233 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 - 17 T + 192 T^{2} - 1615 T^{3} + 8831 T^{4} - 1615 p T^{5} + 192 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.067924828755025971252950313976, −8.288302585936006703225372111740, −8.102721981548189518230283964633, −8.093732273481845721941322941203, −8.017949150503181425676963029325, −7.64941458243845030308416010526, −7.27924133666319223008702019418, −6.75115014833070283486044015618, −6.64677069365387458156733330604, −6.55263229769220659941255111949, −6.08968059892339447994911850108, −5.87858042809533219002491134317, −5.64229514701566240264565907388, −5.16379623109223910251492358918, −4.92481144066176343642685981304, −4.59991917181909407194680261848, −4.35985454269716408791137110575, −4.25965419056545079854625141487, −3.88477580402045776520100038064, −3.24214224253041763523208463276, −3.08692683359994346296652748293, −2.49997133089484997108409607897, −2.44187535877568199674217879546, −1.29418741503686158838866551520, −1.01902304890237170695972190348,
1.01902304890237170695972190348, 1.29418741503686158838866551520, 2.44187535877568199674217879546, 2.49997133089484997108409607897, 3.08692683359994346296652748293, 3.24214224253041763523208463276, 3.88477580402045776520100038064, 4.25965419056545079854625141487, 4.35985454269716408791137110575, 4.59991917181909407194680261848, 4.92481144066176343642685981304, 5.16379623109223910251492358918, 5.64229514701566240264565907388, 5.87858042809533219002491134317, 6.08968059892339447994911850108, 6.55263229769220659941255111949, 6.64677069365387458156733330604, 6.75115014833070283486044015618, 7.27924133666319223008702019418, 7.64941458243845030308416010526, 8.017949150503181425676963029325, 8.093732273481845721941322941203, 8.102721981548189518230283964633, 8.288302585936006703225372111740, 9.067924828755025971252950313976
