| L(s) = 1 | + 2-s − 5-s + 5·9-s − 10-s − 5·11-s + 13-s + 5·18-s − 5·22-s + 26-s − 31-s − 37-s + 41-s + 43-s − 5·45-s − 47-s + 5·49-s − 53-s + 5·55-s − 62-s − 65-s − 67-s − 74-s + 79-s + 15·81-s + 82-s + 83-s + 86-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s + 5·9-s − 10-s − 5·11-s + 13-s + 5·18-s − 5·22-s + 26-s − 31-s − 37-s + 41-s + 43-s − 5·45-s − 47-s + 5·49-s − 53-s + 5·55-s − 62-s − 65-s − 67-s − 74-s + 79-s + 15·81-s + 82-s + 83-s + 86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 149^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 149^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.812186156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812186156\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 + T )^{5} \) |
| 149 | $C_1$ | \( ( 1 + T )^{5} \) |
| good | 2 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 5 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 13 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 31 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 37 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 41 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 43 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 47 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 53 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 67 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 79 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 83 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.86333919393953720712630632781, −5.62797145858596236717231223954, −5.20587205014181923414961669936, −5.16523614008019721986931481547, −5.11513765964475771141380518312, −5.00459954430413315676488460533, −4.54684242912588496141258037579, −4.52226048389039171857772780611, −4.50926597363829723951203857267, −4.50855172360119910455989588297, −3.85533486886069327681159816252, −3.82586986607491225422158413863, −3.78977117293888369663500645147, −3.73007687175564729836919657127, −3.41687932666789020949594025352, −3.10721616634446961088516680114, −2.62503090041648857293157840730, −2.60574990946182001377128292025, −2.50111511526238545857425774303, −2.08574379024668778768221950481, −1.87058245062242554533984707957, −1.82687078923438995194039661081, −1.26586894372916836469215692354, −1.01891704136315221620127410073, −0.67966963927129155411086248780,
0.67966963927129155411086248780, 1.01891704136315221620127410073, 1.26586894372916836469215692354, 1.82687078923438995194039661081, 1.87058245062242554533984707957, 2.08574379024668778768221950481, 2.50111511526238545857425774303, 2.60574990946182001377128292025, 2.62503090041648857293157840730, 3.10721616634446961088516680114, 3.41687932666789020949594025352, 3.73007687175564729836919657127, 3.78977117293888369663500645147, 3.82586986607491225422158413863, 3.85533486886069327681159816252, 4.50855172360119910455989588297, 4.50926597363829723951203857267, 4.52226048389039171857772780611, 4.54684242912588496141258037579, 5.00459954430413315676488460533, 5.11513765964475771141380518312, 5.16523614008019721986931481547, 5.20587205014181923414961669936, 5.62797145858596236717231223954, 5.86333919393953720712630632781
