| L(s) = 1 | − 8·2-s − 18·3-s + 48·4-s − 50·5-s + 144·6-s − 98·7-s − 256·8-s + 243·9-s + 400·10-s + 438·11-s − 864·12-s + 146·13-s + 784·14-s + 900·15-s + 1.28e3·16-s − 656·17-s − 1.94e3·18-s + 1.99e3·19-s − 2.40e3·20-s + 1.76e3·21-s − 3.50e3·22-s + 1.03e3·23-s + 4.60e3·24-s + 1.87e3·25-s − 1.16e3·26-s − 2.91e3·27-s − 4.70e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.26·10-s + 1.09·11-s − 1.73·12-s + 0.239·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.550·17-s − 1.41·18-s + 1.26·19-s − 1.34·20-s + 0.872·21-s − 1.54·22-s + 0.406·23-s + 1.63·24-s + 3/5·25-s − 0.338·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 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 | 11 | $D_{4}$ | \( 1 - 438 T + 253854 T^{2} - 438 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 146 T - 297966 T^{2} - 146 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 656 T + 2482462 T^{2} + 656 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1990 T + 159842 p T^{2} - 1990 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1032 T + 11279598 T^{2} - 1032 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2760 T + 31305798 T^{2} - 2760 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7834 T + 69695966 T^{2} - 7834 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8396 T + 109827518 T^{2} - 8396 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8000 T + 169155118 T^{2} + 8000 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2956 T + 91208694 T^{2} - 2956 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 32036 T + 659021182 T^{2} + 32036 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 59026 T + 1656159986 T^{2} + 59026 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 67436 T + 2543975158 T^{2} + 67436 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 29704 T + 180119750 T^{2} + 29704 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 94332 T + 4719889094 T^{2} + 94332 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 34570 T + 588184678 T^{2} + 34570 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 25490 T - 480278470 T^{2} + 25490 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14552 T - 1172683362 T^{2} - 14552 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 121432 T + 11378579542 T^{2} + 121432 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 91796 T + 12203763158 T^{2} + 91796 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 68054 T + 15168727218 T^{2} + 68054 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.08190593103038755544208323525, −10.97608287146296084436341285478, −10.19095419542074383241173547168, −9.743987794315454596390945198545, −9.393388397244724359591780104561, −8.896299324162586796278340115123, −8.167681415144037510557817988226, −7.80823785695757653279964155473, −7.07203576085284711578752302862, −6.79897121293972092960120889943, −6.11732631490029692674495557225, −5.97886080936414140034150209808, −4.58944461484730459826202809274, −4.55270463720623534635523444849, −3.12758423966450109143966281773, −3.11412187605513505446743628740, −1.31753367047559093415556603870, −1.31280043220093681382978413856, 0, 0,
1.31280043220093681382978413856, 1.31753367047559093415556603870, 3.11412187605513505446743628740, 3.12758423966450109143966281773, 4.55270463720623534635523444849, 4.58944461484730459826202809274, 5.97886080936414140034150209808, 6.11732631490029692674495557225, 6.79897121293972092960120889943, 7.07203576085284711578752302862, 7.80823785695757653279964155473, 8.167681415144037510557817988226, 8.896299324162586796278340115123, 9.393388397244724359591780104561, 9.743987794315454596390945198545, 10.19095419542074383241173547168, 10.97608287146296084436341285478, 11.08190593103038755544208323525
