| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 8·7-s + 5·8-s + 4·11-s − 2·12-s − 4·13-s − 16·14-s + 5·16-s + 2·17-s − 5·19-s + 8·21-s + 8·22-s − 14·23-s − 5·24-s − 8·26-s − 16·28-s − 5·29-s − 7·31-s − 2·32-s − 4·33-s + 4·34-s + 7·37-s − 10·38-s + 4·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 3.02·7-s + 1.76·8-s + 1.20·11-s − 0.577·12-s − 1.10·13-s − 4.27·14-s + 5/4·16-s + 0.485·17-s − 1.14·19-s + 1.74·21-s + 1.70·22-s − 2.91·23-s − 1.02·24-s − 1.56·26-s − 3.02·28-s − 0.928·29-s − 1.25·31-s − 0.353·32-s − 0.696·33-s + 0.685·34-s + 1.15·37-s − 1.62·38-s + 0.640·39-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
| \(L(1)\) |
\(\approx\) |
\(0.1069243306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1069243306\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 50 T^{3} - 299 T^{4} - 50 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 50 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 60 T^{3} + 101 T^{4} + 60 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 5 T + 6 T^{2} - 65 T^{3} - 439 T^{4} - 65 p T^{5} + 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 5 T + 11 T^{2} + 195 T^{3} + 1736 T^{4} + 195 p T^{5} + 11 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 91 p T^{5} + 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 7 T + 32 T^{2} - 365 T^{3} + 3451 T^{4} - 365 p T^{5} + 32 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 17 T + 68 T^{2} - 661 T^{3} - 8425 T^{4} - 661 p T^{5} + 68 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 13 T + p T^{2} + 325 T^{3} + 4016 T^{4} + 325 p T^{5} + p^{3} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_4\times C_2$ | \( 1 + 15 T + 76 T^{2} + 675 T^{3} + 8161 T^{4} + 675 p T^{5} + 76 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 7 T - 12 T^{2} - 511 T^{3} - 2845 T^{4} - 511 p T^{5} - 12 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 21 T + 243 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4\times C_2$ | \( 1 - 8 T - 7 T^{2} + 624 T^{3} - 4495 T^{4} + 624 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - T + 68 T^{2} - 155 T^{3} + 5911 T^{4} - 155 p T^{5} + 68 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 14 T + 53 T^{2} + 870 T^{3} + 14801 T^{4} + 870 p T^{5} + 53 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 15 T + 223 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 17 T + 12 T^{2} + 1445 T^{3} - 15649 T^{4} + 1445 p T^{5} + 12 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.20384645926145525975919806807, −6.91931501483441257493529056157, −6.86054126185474796248598452427, −6.43574634451554850499117830064, −6.40527488888691036143214185177, −6.33248016304381034760768877178, −5.89537099529189710721951981440, −5.72522496033156967759527347341, −5.69456221622130476997759505482, −5.09082857510897851659519564545, −4.83485764315761813061449424522, −4.74153446284264675318447870463, −4.67177656581046734996776570149, −4.07617219243315299821984002140, −3.91554018977793055243220114423, −3.62614497854630957909065112707, −3.46360648370483208050470760115, −3.35122907984893486292600853472, −3.19926261187772944115356045480, −2.30270917107048918211954514523, −2.17225432620334087779331798204, −1.90819776815318837975426766454, −1.85000397378124599004819928640, −0.798466661478890460909903232619, −0.07779185794041522134868326798,
0.07779185794041522134868326798, 0.798466661478890460909903232619, 1.85000397378124599004819928640, 1.90819776815318837975426766454, 2.17225432620334087779331798204, 2.30270917107048918211954514523, 3.19926261187772944115356045480, 3.35122907984893486292600853472, 3.46360648370483208050470760115, 3.62614497854630957909065112707, 3.91554018977793055243220114423, 4.07617219243315299821984002140, 4.67177656581046734996776570149, 4.74153446284264675318447870463, 4.83485764315761813061449424522, 5.09082857510897851659519564545, 5.69456221622130476997759505482, 5.72522496033156967759527347341, 5.89537099529189710721951981440, 6.33248016304381034760768877178, 6.40527488888691036143214185177, 6.43574634451554850499117830064, 6.86054126185474796248598452427, 6.91931501483441257493529056157, 7.20384645926145525975919806807
