| L(s) = 1 | − 13·2-s + 18·3-s + 79·4-s + 50·5-s − 234·6-s − 98·7-s − 273·8-s + 243·9-s − 650·10-s − 392·11-s + 1.42e3·12-s − 676·13-s + 1.27e3·14-s + 900·15-s + 823·16-s − 428·17-s − 3.15e3·18-s + 408·19-s + 3.95e3·20-s − 1.76e3·21-s + 5.09e3·22-s − 2.48e3·23-s − 4.91e3·24-s + 1.87e3·25-s + 8.78e3·26-s + 2.91e3·27-s − 7.74e3·28-s + ⋯ |
| L(s) = 1 | − 2.29·2-s + 1.15·3-s + 2.46·4-s + 0.894·5-s − 2.65·6-s − 0.755·7-s − 1.50·8-s + 9-s − 2.05·10-s − 0.976·11-s + 2.85·12-s − 1.10·13-s + 1.73·14-s + 1.03·15-s + 0.803·16-s − 0.359·17-s − 2.29·18-s + 0.259·19-s + 2.20·20-s − 0.872·21-s + 2.24·22-s − 0.977·23-s − 1.74·24-s + 3/5·25-s + 2.54·26-s + 0.769·27-s − 1.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 13 T + 45 p T^{2} + 13 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 392 T + 293958 T^{2} + 392 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 p^{2} T + 64630 p T^{2} + 4 p^{7} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 428 T + 2881350 T^{2} + 428 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 408 T - 2698026 T^{2} - 408 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2480 T + 7754286 T^{2} + 2480 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 14324 T + 90481982 T^{2} + 14324 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9360 T + 78823742 T^{2} + 9360 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10932 T + 156434510 T^{2} + 10932 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12164 T + 126327126 T^{2} - 12164 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1832 T + 291594502 T^{2} + 1832 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 50960 T + 1103390174 T^{2} + 50960 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 23492 T + 958880142 T^{2} + 23492 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7144 T + 1345226342 T^{2} - 7144 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15036 T + 857785886 T^{2} - 15036 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10808 T + 2520721270 T^{2} + 10808 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5600 T + 2012032142 T^{2} + 5600 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 20692 T - 7256138 T^{2} - 20692 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 43632 T + 4550678814 T^{2} + 43632 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 59144 T + 8524799510 T^{2} - 59144 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 40452 T + 5989161014 T^{2} - 40452 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5436 T + 15704369478 T^{2} + 5436 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.74236247034765095322706666587, −11.97236861577099839821948347949, −10.93023295330734104978166578161, −10.71368562550748639005176472029, −9.813440226200087732984201535751, −9.798803300248903239786393233510, −9.222714524771607325650761099180, −9.138795202476026228362302280388, −8.112828457749242995357789455282, −7.925788823091906989603130420913, −7.28528999823110360282634488070, −6.80164771830179195322329163045, −5.74267133302613628298472258389, −5.13684677838853367279269769280, −3.76364820002915284123967668568, −2.99662095567734882144607910958, −1.95839995178579365637178352554, −1.75591593943336751378309174065, 0, 0,
1.75591593943336751378309174065, 1.95839995178579365637178352554, 2.99662095567734882144607910958, 3.76364820002915284123967668568, 5.13684677838853367279269769280, 5.74267133302613628298472258389, 6.80164771830179195322329163045, 7.28528999823110360282634488070, 7.925788823091906989603130420913, 8.112828457749242995357789455282, 9.138795202476026228362302280388, 9.222714524771607325650761099180, 9.798803300248903239786393233510, 9.813440226200087732984201535751, 10.71368562550748639005176472029, 10.93023295330734104978166578161, 11.97236861577099839821948347949, 12.74236247034765095322706666587
