| L(s) = 1 | + 3·2-s + 6·3-s + 2·4-s + 18·6-s − 3·8-s + 27·9-s + 12·12-s + 4·13-s − 3·16-s − 27·17-s + 81·18-s − 11·19-s − 18·24-s + 38·25-s + 12·26-s + 108·27-s + 78·29-s − 32·31-s − 12·32-s − 81·34-s + 54·36-s + 34·37-s − 33·38-s + 24·39-s − 21·41-s + 61·43-s + 84·47-s + ⋯ |
| L(s) = 1 | + 3/2·2-s + 2·3-s + 1/2·4-s + 3·6-s − 3/8·8-s + 3·9-s + 12-s + 4/13·13-s − 0.187·16-s − 1.58·17-s + 9/2·18-s − 0.578·19-s − 3/4·24-s + 1.51·25-s + 6/13·26-s + 4·27-s + 2.68·29-s − 1.03·31-s − 3/8·32-s − 2.38·34-s + 3/2·36-s + 0.918·37-s − 0.868·38-s + 8/13·39-s − 0.512·41-s + 1.41·43-s + 1.78·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.94894767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.94894767\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 38 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 239 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T - 153 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 27 T + 532 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 32 T + 63 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 21 T + 1828 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 84 T + 4561 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 87 T + 6004 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 56 T - 585 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 65 T - 1104 T^{2} + 65 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 216 T + 23473 T^{2} - 216 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.88412713466538810920506194542, −10.69000337665265712285985069629, −10.41390046773913804259601474445, −9.595204439592862508166623476297, −9.066159720185176860497601292033, −8.957398349723348297134739665787, −8.478287867491043460030038424428, −8.055842192656439783950212172647, −7.33927356908189366838490077343, −7.03816142807988026512299499909, −6.41026247929694589950661301747, −5.99312531683457480015699663656, −4.81162938268634913473921831495, −4.79802228635547893216331992136, −4.14584942411494333995207454940, −3.93911997160820473683707040781, −2.98572404861385715991670418901, −2.79794225313774067029137272078, −2.05347084757411291618423909921, −1.06559652123727098244201505184,
1.06559652123727098244201505184, 2.05347084757411291618423909921, 2.79794225313774067029137272078, 2.98572404861385715991670418901, 3.93911997160820473683707040781, 4.14584942411494333995207454940, 4.79802228635547893216331992136, 4.81162938268634913473921831495, 5.99312531683457480015699663656, 6.41026247929694589950661301747, 7.03816142807988026512299499909, 7.33927356908189366838490077343, 8.055842192656439783950212172647, 8.478287867491043460030038424428, 8.957398349723348297134739665787, 9.066159720185176860497601292033, 9.595204439592862508166623476297, 10.41390046773913804259601474445, 10.69000337665265712285985069629, 10.88412713466538810920506194542
