| L(s) = 1 | − 14·9-s − 76·13-s − 68·17-s + 540·29-s − 412·37-s − 540·41-s − 326·49-s + 516·53-s − 500·61-s + 2.15e3·73-s − 533·81-s + 1.78e3·89-s + 508·97-s + 1.19e3·101-s + 1.70e3·109-s − 3.39e3·113-s + 1.06e3·117-s − 2.50e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 952·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 0.518·9-s − 1.62·13-s − 0.970·17-s + 3.45·29-s − 1.83·37-s − 2.05·41-s − 0.950·49-s + 1.33·53-s − 1.04·61-s + 3.45·73-s − 0.731·81-s + 2.11·89-s + 0.531·97-s + 1.17·101-s + 1.50·109-s − 2.82·113-s + 0.840·117-s − 1.87·121-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 + 0.503·153-s + 0.000508·157-s + 0.000480·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9801363152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9801363152\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 14 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 326 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2502 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 3478 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 17574 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 270 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 57058 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 206 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 129986 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 190006 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 258 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 404998 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 250 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 64114 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 299662 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1078 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 908638 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 81426 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 p T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
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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
−10.23451933887939789669805235536, −9.701997660618840758960782394623, −9.131569080247708413189564627381, −8.900962433995662839984440889367, −8.231376333426647150968870971040, −8.179912953935986957600017457681, −7.55056901872421358901105903769, −6.84315727494272367078024195662, −6.60619216345694006662681261185, −6.49552164182341887712946692670, −5.46446838164856402271630170695, −5.17645615559185625847641613917, −4.69133480429537558933889286481, −4.43395999108753006251037357154, −3.43643670057893878187351628298, −3.15688626071505511407028331992, −2.35348824279759641073464597163, −2.11820551253215544693149834386, −1.09796248706382642042406981247, −0.28523851105051543404136060878,
0.28523851105051543404136060878, 1.09796248706382642042406981247, 2.11820551253215544693149834386, 2.35348824279759641073464597163, 3.15688626071505511407028331992, 3.43643670057893878187351628298, 4.43395999108753006251037357154, 4.69133480429537558933889286481, 5.17645615559185625847641613917, 5.46446838164856402271630170695, 6.49552164182341887712946692670, 6.60619216345694006662681261185, 6.84315727494272367078024195662, 7.55056901872421358901105903769, 8.179912953935986957600017457681, 8.231376333426647150968870971040, 8.900962433995662839984440889367, 9.131569080247708413189564627381, 9.701997660618840758960782394623, 10.23451933887939789669805235536
