L(s) = 1 | + 32·7-s − 9·9-s − 252·17-s − 336·23-s + 214·25-s − 176·31-s − 84·41-s − 192·47-s + 82·49-s − 288·63-s − 1.58e3·71-s − 436·73-s − 1.04e3·79-s + 81·81-s − 1.62e3·89-s + 2.30e3·97-s − 256·103-s − 924·113-s − 8.06e3·119-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.26e3·153-s + ⋯ |
L(s) = 1 | + 1.72·7-s − 1/3·9-s − 3.59·17-s − 3.04·23-s + 1.71·25-s − 1.01·31-s − 0.319·41-s − 0.595·47-s + 0.239·49-s − 0.575·63-s − 2.64·71-s − 0.699·73-s − 1.48·79-s + 1/9·81-s − 1.92·89-s + 2.41·97-s − 0.244·103-s − 0.769·113-s − 6.21·119-s + 1.89·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 1.19·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 214 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2518 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13318 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 47878 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156310 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 258550 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 24842 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 164518 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 179930 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 792 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 218 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 520 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 901510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1154 T + p^{3} T^{2} )^{2} \) |
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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
−9.728764591749159091974519878510, −9.045355283944169556423918986879, −8.701533324920314270409308117076, −8.635527349977539747011059829983, −8.022020101694580846596646665026, −7.71842667830765278883420758693, −6.98699398808047146551979605127, −6.79607051684719350040688854980, −6.04759582014127750420073460243, −5.89490957806419267866056687711, −4.92668916845336885518751298676, −4.80812291537779864572969360517, −4.22693213939216774204977394023, −4.00223952033311384578301692134, −2.96440634633205244086595789393, −2.32314795007014980898382008473, −1.86583473427032091233652659999, −1.47882309135144815357803456528, 0, 0,
1.47882309135144815357803456528, 1.86583473427032091233652659999, 2.32314795007014980898382008473, 2.96440634633205244086595789393, 4.00223952033311384578301692134, 4.22693213939216774204977394023, 4.80812291537779864572969360517, 4.92668916845336885518751298676, 5.89490957806419267866056687711, 6.04759582014127750420073460243, 6.79607051684719350040688854980, 6.98699398808047146551979605127, 7.71842667830765278883420758693, 8.022020101694580846596646665026, 8.635527349977539747011059829983, 8.701533324920314270409308117076, 9.045355283944169556423918986879, 9.728764591749159091974519878510