| L(s) = 1 | + 5-s + 2·7-s − 2·11-s − 13-s − 6·17-s − 4·19-s − 6·23-s + 5·25-s + 29-s − 8·31-s + 2·35-s + 2·37-s − 2·41-s + 10·43-s − 4·47-s + 7·49-s + 20·53-s − 2·55-s − 4·59-s − 9·61-s − 65-s − 14·67-s + 20·71-s − 18·73-s − 4·77-s + 10·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.603·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s − 1.25·23-s + 25-s + 0.185·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s − 0.312·41-s + 1.52·43-s − 0.583·47-s + 49-s + 2.74·53-s − 0.269·55-s − 0.520·59-s − 1.15·61-s − 0.124·65-s − 1.71·67-s + 2.37·71-s − 2.10·73-s − 0.455·77-s + 1.12·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.467439406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.467439406\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.911255708781344729933663402176, −8.630154931725456396082664652481, −8.554807035496534996533316220208, −7.83675023880281529294798967000, −7.62108880815641104670994543659, −7.17320173651545764709100144144, −6.80271395109578091318660837622, −6.43018109204937428523431534766, −5.83874839468472582514060502831, −5.62301093562828267718676625827, −5.28442593520357941364323442290, −4.50693775624085513391284015302, −4.44554280208343950487755281782, −4.07703912151907414651163869830, −3.38574537237918447587381315370, −2.69544636619853081774063136119, −2.33832156473046897884886547834, −1.97459358961006231662105838409, −1.37365606148837335769490840013, −0.38776380686161852716118348098,
0.38776380686161852716118348098, 1.37365606148837335769490840013, 1.97459358961006231662105838409, 2.33832156473046897884886547834, 2.69544636619853081774063136119, 3.38574537237918447587381315370, 4.07703912151907414651163869830, 4.44554280208343950487755281782, 4.50693775624085513391284015302, 5.28442593520357941364323442290, 5.62301093562828267718676625827, 5.83874839468472582514060502831, 6.43018109204937428523431534766, 6.80271395109578091318660837622, 7.17320173651545764709100144144, 7.62108880815641104670994543659, 7.83675023880281529294798967000, 8.554807035496534996533316220208, 8.630154931725456396082664652481, 8.911255708781344729933663402176
