| L(s) = 1 | + 29·9-s + 30·11-s + 210·19-s + 40·29-s + 460·31-s − 390·41-s + 586·49-s + 1.12e3·59-s − 1.46e3·61-s + 80·71-s − 1.66e3·79-s + 112·81-s + 1.41e3·89-s + 870·99-s − 3.80e3·101-s + 492·109-s − 1.98e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.33e3·169-s + ⋯ |
| L(s) = 1 | + 1.07·9-s + 0.822·11-s + 2.53·19-s + 0.256·29-s + 2.66·31-s − 1.48·41-s + 1.70·49-s + 2.47·59-s − 3.06·61-s + 0.133·71-s − 2.36·79-s + 0.153·81-s + 1.67·89-s + 0.883·99-s − 3.74·101-s + 0.432·109-s − 1.49·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.97·169-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
| \(L(2)\) |
\(\approx\) |
\(4.538801186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.538801186\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 586 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 15 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4330 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9385 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 105 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24234 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 98390 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 195 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 69014 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22754 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 194070 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 560 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 730 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 536501 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 40 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 677545 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 830 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1137949 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 705 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 231010 T^{2} + p^{6} T^{4} \) |
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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
−10.05933523694180690769316812377, −9.716079964802622421143196220288, −9.390462472179069596155106907535, −8.695014365901603831648621809636, −8.588597613853944861855092057133, −7.70072005272500346146200503196, −7.63561133962991080667685581651, −7.06974186968484141758342007936, −6.60857381789066001051997728776, −6.33301783298817688197392994370, −5.58153553336405977469609779422, −5.18653088557902217491367584193, −4.71570131924286179387925384088, −4.07163811184688328759868427161, −3.82731252784844201825549473418, −2.88791181688403196196396084694, −2.78126844935986425972026293445, −1.54790423162024687556825623203, −1.26488385555692970705924787355, −0.64613833073178231131757435988,
0.64613833073178231131757435988, 1.26488385555692970705924787355, 1.54790423162024687556825623203, 2.78126844935986425972026293445, 2.88791181688403196196396084694, 3.82731252784844201825549473418, 4.07163811184688328759868427161, 4.71570131924286179387925384088, 5.18653088557902217491367584193, 5.58153553336405977469609779422, 6.33301783298817688197392994370, 6.60857381789066001051997728776, 7.06974186968484141758342007936, 7.63561133962991080667685581651, 7.70072005272500346146200503196, 8.588597613853944861855092057133, 8.695014365901603831648621809636, 9.390462472179069596155106907535, 9.716079964802622421143196220288, 10.05933523694180690769316812377
