| 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
−11.59955754166746515546209313645, −10.59183791980375634153117254510, −9.803394309413774688439520628373, −8.298927818883088908865609383543, −7.33459421101593872145323112438, −6.75584192827493098425875746785, −5.13213278536282295641004715602, −3.84390582812221012358042187773, −2.06356357948349134028007177809, −0.46670368239130282007671563142,
2.18867434873654085315857788228, 3.64721020269240167102068460003, 5.09020322343278969978702017998, 5.84378088159192801725104328970, 7.47830241340279946549230284307, 8.577302164423857226074105349549, 9.505128745217026557351761796753, 10.44475585085297730829102026761, 11.26069876463917654328974692881, 12.63675839951850920524275127520
