| L(s) = 1 | − 48·7-s − 9·9-s + 164·17-s + 16·23-s + 54·25-s − 160·31-s − 564·41-s − 480·47-s + 1.04e3·49-s + 432·63-s + 1.71e3·71-s + 1.99e3·73-s + 64·79-s + 81·81-s + 492·89-s + 1.73e3·97-s − 2.99e3·103-s + 1.57e3·113-s − 7.87e3·119-s + 1.87e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.47e3·153-s + ⋯ |
| L(s) = 1 | − 2.59·7-s − 1/3·9-s + 2.33·17-s + 0.145·23-s + 0.431·25-s − 0.926·31-s − 2.14·41-s − 1.48·47-s + 3.03·49-s + 0.863·63-s + 2.86·71-s + 3.20·73-s + 0.0911·79-s + 1/9·81-s + 0.585·89-s + 1.81·97-s − 2.86·103-s + 1.30·113-s − 6.06·119-s + 1.41·121-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 − 0.779·153-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
| \(L(2)\) |
\(\approx\) |
\(1.060659915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060659915\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 54 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 1878 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 1082 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 5254 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 29734 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 80 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 282 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158998 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 280854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 55542 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 406438 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 411430 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 856 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 998 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1130490 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 866 T + p^{3} T^{2} )^{2} \) |
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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
−10.19163377475394451099004904742, −9.843096620261683063123788637639, −9.357980000937952687519437417531, −9.062189721558620557559479501246, −8.468018738034087619685244699889, −7.83562745865748635318416457442, −7.73507435079042067658638917298, −6.76412950236055971932347983173, −6.71365559187858951590043620734, −6.36488920915821683270876777282, −5.74551550750571376064039165237, −5.15055055744073820492781780992, −5.09051760507480844054882651891, −3.73665523670741735984778100129, −3.61171907099952536160179163117, −3.25465306421631206924755681275, −2.76678432207595766815678875213, −1.92615398707818483023844590106, −0.984356794191576457958615555067, −0.33227264278853253182183223493,
0.33227264278853253182183223493, 0.984356794191576457958615555067, 1.92615398707818483023844590106, 2.76678432207595766815678875213, 3.25465306421631206924755681275, 3.61171907099952536160179163117, 3.73665523670741735984778100129, 5.09051760507480844054882651891, 5.15055055744073820492781780992, 5.74551550750571376064039165237, 6.36488920915821683270876777282, 6.71365559187858951590043620734, 6.76412950236055971932347983173, 7.73507435079042067658638917298, 7.83562745865748635318416457442, 8.468018738034087619685244699889, 9.062189721558620557559479501246, 9.357980000937952687519437417531, 9.843096620261683063123788637639, 10.19163377475394451099004904742
