| L(s) = 1 | + 4i·3-s + 24i·7-s + 11·9-s − 44·11-s − 22i·13-s + 50i·17-s − 44·19-s − 96·21-s + 56i·23-s + 152i·27-s − 198·29-s − 160·31-s − 176i·33-s − 162i·37-s + 88·39-s + ⋯ |
| L(s) = 1 | + 0.769i·3-s + 1.29i·7-s + 0.407·9-s − 1.20·11-s − 0.469i·13-s + 0.713i·17-s − 0.531·19-s − 0.997·21-s + 0.507i·23-s + 1.08i·27-s − 1.26·29-s − 0.926·31-s − 0.928i·33-s − 0.719i·37-s + 0.361·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.258927 + 1.09683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.258927 + 1.09683i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4iT - 27T^{2} \) |
| 7 | \( 1 - 24iT - 343T^{2} \) |
| 11 | \( 1 + 44T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 50iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 44T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 198T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 198T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 528iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 242iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 668T + 2.05e5T^{2} \) |
| 61 | \( 1 - 550T + 2.26e5T^{2} \) |
| 67 | \( 1 - 188iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 728T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 656T + 4.93e5T^{2} \) |
| 83 | \( 1 + 236iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 714T + 7.04e5T^{2} \) |
| 97 | \( 1 + 478iT - 9.12e5T^{2} \) |
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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
−12.63675839951850920524275127520, −11.26069876463917654328974692881, −10.44475585085297730829102026761, −9.505128745217026557351761796753, −8.577302164423857226074105349549, −7.47830241340279946549230284307, −5.84378088159192801725104328970, −5.09020322343278969978702017998, −3.64721020269240167102068460003, −2.18867434873654085315857788228,
0.46670368239130282007671563142, 2.06356357948349134028007177809, 3.84390582812221012358042187773, 5.13213278536282295641004715602, 6.75584192827493098425875746785, 7.33459421101593872145323112438, 8.298927818883088908865609383543, 9.803394309413774688439520628373, 10.59183791980375634153117254510, 11.59955754166746515546209313645
