| L(s) = 1 | − 5i·3-s − 10i·7-s + 2·9-s + 15·11-s − 8i·13-s − 21i·17-s + 105·19-s − 50·21-s + 10i·23-s − 145i·27-s + 20·29-s + 230·31-s − 75i·33-s − 54i·37-s − 40·39-s + ⋯ |
| L(s) = 1 | − 0.962i·3-s − 0.539i·7-s + 0.0740·9-s + 0.411·11-s − 0.170i·13-s − 0.299i·17-s + 1.26·19-s − 0.519·21-s + 0.0906i·23-s − 1.03i·27-s + 0.128·29-s + 1.33·31-s − 0.395i·33-s − 0.239i·37-s − 0.164·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.130446241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130446241\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5iT - 27T^{2} \) |
| 7 | \( 1 + 10iT - 343T^{2} \) |
| 11 | \( 1 - 15T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 21iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 105T + 6.85e3T^{2} \) |
| 23 | \( 1 - 10iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 20T + 2.43e4T^{2} \) |
| 31 | \( 1 - 230T + 2.97e4T^{2} \) |
| 37 | \( 1 + 54iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 195T + 6.89e4T^{2} \) |
| 43 | \( 1 - 300iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 480iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 322iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 560T + 2.05e5T^{2} \) |
| 61 | \( 1 + 730T + 2.26e5T^{2} \) |
| 67 | \( 1 - 255iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 40T + 3.57e5T^{2} \) |
| 73 | \( 1 + 317iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 830T + 4.93e5T^{2} \) |
| 83 | \( 1 + 75iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 705T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.716079964802622421143196220288, −8.588597613853944861855092057133, −7.70072005272500346146200503196, −7.06974186968484141758342007936, −6.33301783298817688197392994370, −5.18653088557902217491367584193, −4.07163811184688328759868427161, −2.88791181688403196196396084694, −1.54790423162024687556825623203, −0.64613833073178231131757435988,
1.26488385555692970705924787355, 2.78126844935986425972026293445, 3.82731252784844201825549473418, 4.71570131924286179387925384088, 5.58153553336405977469609779422, 6.60857381789066001051997728776, 7.63561133962991080667685581651, 8.695014365901603831648621809636, 9.390462472179069596155106907535, 10.05933523694180690769316812377
