| L(s) = 1 | + 2-s − 3-s + 5·5-s − 6-s − 12·7-s − 2·9-s + 5·10-s + 3·11-s − 13-s − 12·14-s − 5·15-s + 3·17-s − 2·18-s − 10·19-s + 12·21-s + 3·22-s + 9·23-s + 10·25-s − 26-s + 15·29-s − 5·30-s + 3·31-s − 32-s − 3·33-s + 3·34-s − 60·35-s − 17·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 2.23·5-s − 0.408·6-s − 4.53·7-s − 2/3·9-s + 1.58·10-s + 0.904·11-s − 0.277·13-s − 3.20·14-s − 1.29·15-s + 0.727·17-s − 0.471·18-s − 2.29·19-s + 2.61·21-s + 0.639·22-s + 1.87·23-s + 2·25-s − 0.196·26-s + 2.78·29-s − 0.912·30-s + 0.538·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s − 10.1·35-s − 2.79·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6250000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5186484406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5186484406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + T + p T^{2} + 5 T^{3} + 16 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 3 T + 8 T^{2} - 51 T^{3} + 265 T^{4} - 51 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 - 3 T + 37 T^{2} - 45 T^{3} + 676 T^{4} - 45 p T^{5} + 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 9 T + 8 T^{2} + 195 T^{3} - 1259 T^{4} + 195 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 15 T + 56 T^{2} + 315 T^{3} - 3629 T^{4} + 315 p T^{5} + 56 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 3 T - 12 T^{2} - 131 T^{3} + 1365 T^{4} - 131 p T^{5} - 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 17 T + 147 T^{2} + 1135 T^{3} + 7976 T^{4} + 1135 p T^{5} + 147 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 13 T + 28 T^{2} + 169 T^{3} - 945 T^{4} + 169 p T^{5} + 28 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 23 T + 202 T^{2} - 925 T^{3} + 4101 T^{4} - 925 p T^{5} + 202 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 16 T + 43 T^{2} - 700 T^{3} - 7959 T^{4} - 700 p T^{5} + 43 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 10 T + T^{2} + 200 T^{3} + 5061 T^{4} + 200 p T^{5} + p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 13 T + 12 T^{2} + 895 T^{3} - 9919 T^{4} + 895 p T^{5} + 12 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 850 p T^{5} + 63 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 10 T - 39 T^{2} + 10 p T^{3} - 2839 T^{4} + 10 p^{2} T^{5} - 39 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + T + 58 T^{2} + 335 T^{3} + 7041 T^{4} + 335 p T^{5} + 58 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 10 T - 29 T^{2} - 200 T^{3} + 10101 T^{4} - 200 p T^{5} - 29 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 22 T + 207 T^{2} + 2420 T^{3} + 31541 T^{4} + 2420 p T^{5} + 207 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−12.19592463867874050084188239272, −11.35968793571479212922882596834, −10.76405540278108253356939906848, −10.71855054685868898030176639406, −10.47219269246760589905403517313, −9.943503721828066556647466439426, −9.611911275214180421946032340747, −9.610167065681690100723965142551, −9.579639496543400602048680518587, −8.878913600948920218892311061489, −8.663096103605671259099197429464, −8.453699052979076533919704021011, −7.41060135258827907074631692885, −6.82423024056269063702346307207, −6.61173167876963624423268336093, −6.58734495133729244107940252311, −6.13516275761063215096646989342, −6.03090493458654367592850577096, −5.70769077123892510798697730350, −4.95716458485699832699612766014, −4.70857775135264936403522717060, −3.63875109505873099441722556981, −3.43402882809268056309322477990, −2.90410122232110540964064154323, −2.40102396990203361455319952814,
2.40102396990203361455319952814, 2.90410122232110540964064154323, 3.43402882809268056309322477990, 3.63875109505873099441722556981, 4.70857775135264936403522717060, 4.95716458485699832699612766014, 5.70769077123892510798697730350, 6.03090493458654367592850577096, 6.13516275761063215096646989342, 6.58734495133729244107940252311, 6.61173167876963624423268336093, 6.82423024056269063702346307207, 7.41060135258827907074631692885, 8.453699052979076533919704021011, 8.663096103605671259099197429464, 8.878913600948920218892311061489, 9.579639496543400602048680518587, 9.610167065681690100723965142551, 9.611911275214180421946032340747, 9.943503721828066556647466439426, 10.47219269246760589905403517313, 10.71855054685868898030176639406, 10.76405540278108253356939906848, 11.35968793571479212922882596834, 12.19592463867874050084188239272
