| L(s) = 1 | + 4-s − 2·5-s − 6·7-s + 9-s + 2·11-s − 4·13-s − 12·17-s + 6·19-s − 2·20-s − 18·23-s + 5·25-s − 6·28-s − 6·29-s + 16·31-s + 12·35-s + 36-s + 24·37-s + 8·41-s + 6·43-s + 2·44-s − 2·45-s + 8·49-s − 4·52-s − 4·55-s − 4·59-s − 4·61-s − 6·63-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 2.26·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 2.91·17-s + 1.37·19-s − 0.447·20-s − 3.75·23-s + 25-s − 1.13·28-s − 1.11·29-s + 2.87·31-s + 2.02·35-s + 1/6·36-s + 3.94·37-s + 1.24·41-s + 0.914·43-s + 0.301·44-s − 0.298·45-s + 8/7·49-s − 0.554·52-s − 0.539·55-s − 0.520·59-s − 0.512·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3518681848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3518681848\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 93 T^{2} + 540 T^{3} + 2552 T^{4} + 540 p T^{5} + 93 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 18 T + 180 T^{2} + 1296 T^{3} + 7139 T^{4} + 1296 p T^{5} + 180 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T - 19 T^{2} - 18 T^{3} + 1140 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 24 T + 313 T^{2} - 2904 T^{3} + 20376 T^{4} - 2904 p T^{5} + 313 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 8 T - 7 T^{2} + 88 T^{3} + 736 T^{4} + 88 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 76 T^{2} - 384 T^{3} + 2763 T^{4} - 384 p T^{5} + 76 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 4442 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 9675 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 4 T + 2 T^{2} - 416 T^{3} - 4229 T^{4} - 416 p T^{5} + 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 37 T^{2} - 572 T^{3} - 4256 T^{4} - 572 p T^{5} + 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 30 T + 508 T^{2} - 6240 T^{3} + 58875 T^{4} - 6240 p T^{5} + 508 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 112 T^{2} + 36 T^{3} + 14307 T^{4} + 36 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 8883 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 26186 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 4 T - 118 T^{2} + 176 T^{3} + 8611 T^{4} + 176 p T^{5} - 118 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.941602611188863900258747024507, −7.88367739601194536375650332229, −7.78052009526588697359879109846, −7.61766055345221705203669269875, −6.98525166931229021444036387405, −6.89550236291627600852509452525, −6.70879393164166687809352365862, −6.59210139555035814903822577831, −6.08685642104320815327997419468, −6.02405583235058251375757870683, −5.95742327388418673789644972460, −5.56897234690652801146154211269, −4.90146886620983745651467393911, −4.69632397223621869030439562645, −4.32512003416717583439887493534, −4.09332063889690434910328578947, −3.96186596463694873454989311512, −3.91309089736109103429578290090, −3.07732899525056759334105663675, −2.76935609844392191654982738398, −2.57326531118430532999640472499, −2.52814888143233867512910702686, −1.84627632914378257901387473687, −1.09484521836532514058353526637, −0.24802731135425498525498585383,
0.24802731135425498525498585383, 1.09484521836532514058353526637, 1.84627632914378257901387473687, 2.52814888143233867512910702686, 2.57326531118430532999640472499, 2.76935609844392191654982738398, 3.07732899525056759334105663675, 3.91309089736109103429578290090, 3.96186596463694873454989311512, 4.09332063889690434910328578947, 4.32512003416717583439887493534, 4.69632397223621869030439562645, 4.90146886620983745651467393911, 5.56897234690652801146154211269, 5.95742327388418673789644972460, 6.02405583235058251375757870683, 6.08685642104320815327997419468, 6.59210139555035814903822577831, 6.70879393164166687809352365862, 6.89550236291627600852509452525, 6.98525166931229021444036387405, 7.61766055345221705203669269875, 7.78052009526588697359879109846, 7.88367739601194536375650332229, 7.941602611188863900258747024507
