| L(s) = 1 | + 5·2-s − 6·3-s + 12·4-s + 2·5-s − 30·6-s + 7-s + 20·8-s + 23·9-s + 10·10-s − 72·12-s + 6·13-s + 5·14-s − 12·15-s + 30·16-s − 6·17-s + 115·18-s − 8·19-s + 24·20-s − 6·21-s − 8·23-s − 120·24-s + 5·25-s + 30·26-s − 70·27-s + 12·28-s − 6·29-s − 60·30-s + ⋯ |
| L(s) = 1 | + 3.53·2-s − 3.46·3-s + 6·4-s + 0.894·5-s − 12.2·6-s + 0.377·7-s + 7.07·8-s + 23/3·9-s + 3.16·10-s − 20.7·12-s + 1.66·13-s + 1.33·14-s − 3.09·15-s + 15/2·16-s − 1.45·17-s + 27.1·18-s − 1.83·19-s + 5.36·20-s − 1.30·21-s − 1.66·23-s − 24.4·24-s + 25-s + 5.88·26-s − 13.4·27-s + 2.26·28-s − 1.11·29-s − 10.9·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045607388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045607388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 25 p T^{5} + 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 + 2 p T + 13 T^{2} + 10 T^{3} + T^{4} + 10 p T^{5} + 13 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 12 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 + 6 T - T^{2} - 18 T^{3} + 169 T^{4} - 18 p T^{5} - p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 8 T + 45 T^{2} + 268 T^{3} + 1529 T^{4} + 268 p T^{5} + 45 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 6 T + 47 T^{2} + 288 T^{3} + 2365 T^{4} + 288 p T^{5} + 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 + 6 T + 39 T^{2} + 352 T^{3} + 3309 T^{4} + 352 p T^{5} + 39 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 4 T + 55 T^{2} + 326 T^{3} + 1389 T^{4} + 326 p T^{5} + 55 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 10 T - 7 T^{2} - 10 p T^{3} - 3031 T^{4} - 10 p^{2} T^{5} - 7 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} - 480 T^{3} + 5581 T^{4} - 480 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 6 T - 43 T^{2} - 102 T^{3} + 3025 T^{4} - 102 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 10 T - 21 T^{2} - 10 p T^{3} - 3199 T^{4} - 10 p^{2} T^{5} - 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 16 T + 25 T^{2} + 916 T^{3} - 9471 T^{4} + 916 p T^{5} + 25 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 8 T - 49 T^{2} - 446 T^{3} + 1549 T^{4} - 446 p T^{5} - 49 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 20 T + 161 T^{2} + 1600 T^{3} + 18961 T^{4} + 1600 p T^{5} + 161 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 28 T + 461 T^{2} + 5964 T^{3} + 61769 T^{4} + 5964 p T^{5} + 461 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 34 T + 459 T^{2} + 3488 T^{3} + 25829 T^{4} + 3488 p T^{5} + 459 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.77086216135483282032309102345, −6.66740717151559082144934949971, −6.61553370881391553022207552309, −6.49117133187311209684268820257, −6.44588502510846930191913397605, −5.85235031212354717333313489305, −5.84091806396628151621408007185, −5.48842692471518646181184967662, −5.39836702441688353034468488193, −5.23840742409575886587554999579, −4.99657502041282827163756933065, −4.88895912166704354060348468514, −4.52799714307787666467962617319, −4.24763590444809608505626086433, −4.15823573132153968983829558378, −4.00995741872457724875993113021, −3.80431242425887039774423735250, −3.35555453974824048528325694430, −3.23153222755118885349531882657, −2.46638243204561322734470763753, −1.91747948190222301140323438717, −1.85196264082905807521776008050, −1.52776280502114362850400187328, −1.32000787275111845634871855860, −0.15739044998961855801245141137,
0.15739044998961855801245141137, 1.32000787275111845634871855860, 1.52776280502114362850400187328, 1.85196264082905807521776008050, 1.91747948190222301140323438717, 2.46638243204561322734470763753, 3.23153222755118885349531882657, 3.35555453974824048528325694430, 3.80431242425887039774423735250, 4.00995741872457724875993113021, 4.15823573132153968983829558378, 4.24763590444809608505626086433, 4.52799714307787666467962617319, 4.88895912166704354060348468514, 4.99657502041282827163756933065, 5.23840742409575886587554999579, 5.39836702441688353034468488193, 5.48842692471518646181184967662, 5.84091806396628151621408007185, 5.85235031212354717333313489305, 6.44588502510846930191913397605, 6.49117133187311209684268820257, 6.61553370881391553022207552309, 6.66740717151559082144934949971, 6.77086216135483282032309102345
