| L(s) = 1 | + 2-s + 2·4-s − 5·5-s − 5·10-s + 6·11-s − 2·13-s + 8·17-s − 4·19-s − 10·20-s + 6·22-s − 10·23-s + 10·25-s − 2·26-s + 8·29-s − 11·32-s + 8·34-s − 15·37-s − 4·38-s + 4·43-s + 12·44-s − 10·46-s + 2·47-s + 12·49-s + 10·50-s − 4·52-s + 5·53-s − 30·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 2.23·5-s − 1.58·10-s + 1.80·11-s − 0.554·13-s + 1.94·17-s − 0.917·19-s − 2.23·20-s + 1.27·22-s − 2.08·23-s + 2·25-s − 0.392·26-s + 1.48·29-s − 1.94·32-s + 1.37·34-s − 2.46·37-s − 0.648·38-s + 0.609·43-s + 1.80·44-s − 1.47·46-s + 0.291·47-s + 12/7·49-s + 1.41·50-s − 0.554·52-s + 0.686·53-s − 4.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.529367283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529367283\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 6 T + 5 T^{2} + 6 T^{3} + 49 T^{4} + 6 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 2 T + 11 T^{2} + 16 T^{3} + 49 T^{4} + 16 p T^{5} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 8 T + p T^{2} - 60 T^{3} + 461 T^{4} - 60 p T^{5} + p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 4 T - 3 T^{2} + 62 T^{3} + 605 T^{4} + 62 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 200 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 8 T + 5 T^{2} + 8 p T^{3} - 59 p T^{4} + 8 p^{2} T^{5} + 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 9 T^{2} - 110 T^{3} + 741 T^{4} - 110 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 15 T + 63 T^{2} + 65 T^{3} + 144 T^{4} + 65 p T^{5} + 63 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 31 T^{2} - 180 T^{3} + 1501 T^{4} - 180 p T^{5} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 2 T - 23 T^{2} - 250 T^{3} + 2601 T^{4} - 250 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 5 T - 43 T^{2} + 5 p T^{3} + 1244 T^{4} + 5 p^{2} T^{5} - 43 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 4 T - 43 T^{2} - 408 T^{3} + 905 T^{4} - 408 p T^{5} - 43 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 2 T - 57 T^{2} - 64 T^{3} + 3905 T^{4} - 64 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 2 T - 43 T^{2} - 370 T^{3} + 5041 T^{4} - 370 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 8 T - 47 T^{2} - 434 T^{3} + 1365 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 10 T + 27 T^{2} + 740 T^{3} + 11429 T^{4} + 740 p T^{5} + 27 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 18 T + 41 T^{2} + 1416 T^{3} - 18731 T^{4} + 1416 p T^{5} + 41 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 9 T - 43 T^{2} + 993 T^{3} - 4460 T^{4} + 993 p T^{5} - 43 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 2 T - 33 T^{2} - 820 T^{3} + 10061 T^{4} - 820 p T^{5} - 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.860013480804966148367741146245, −8.661811948890745460984599028936, −8.349707992381493234343423797379, −8.058179543206038949223117324633, −7.939672098972339638103391939996, −7.42790405120259009925472471808, −7.39078886440707774786923270002, −7.17897739639323865238545785313, −6.75589357606135995435645406173, −6.64369002915204428445898605805, −6.17268030779864550328744546749, −5.95612910094791863362456194340, −5.70953934062721895829433189962, −5.39051016927796325163416647458, −4.84822832189948904650503146037, −4.71824789779198342601064649634, −4.08437446134245513321465711808, −3.85292289981108673409549077403, −3.83061923643396469933616793890, −3.62989396027649384536275826280, −2.92785447095874564984542966191, −2.86701938106776856420203548953, −1.93949826677312113901839068846, −1.72749929965010951357273397756, −0.63272342484607830176753672313,
0.63272342484607830176753672313, 1.72749929965010951357273397756, 1.93949826677312113901839068846, 2.86701938106776856420203548953, 2.92785447095874564984542966191, 3.62989396027649384536275826280, 3.83061923643396469933616793890, 3.85292289981108673409549077403, 4.08437446134245513321465711808, 4.71824789779198342601064649634, 4.84822832189948904650503146037, 5.39051016927796325163416647458, 5.70953934062721895829433189962, 5.95612910094791863362456194340, 6.17268030779864550328744546749, 6.64369002915204428445898605805, 6.75589357606135995435645406173, 7.17897739639323865238545785313, 7.39078886440707774786923270002, 7.42790405120259009925472471808, 7.939672098972339638103391939996, 8.058179543206038949223117324633, 8.349707992381493234343423797379, 8.661811948890745460984599028936, 8.860013480804966148367741146245
