| L(s) = 1 | + 2.23·3-s − 5-s − 9.79i·7-s − 3.99·9-s − 6.70·11-s − 5·13-s − 2.23·15-s − 21.9i·17-s − 13.4·19-s − 21.9i·21-s + 9.79i·23-s − 24·25-s − 29.0·27-s + (−19 + 21.9i)29-s + 46.9·31-s + ⋯ |
| L(s) = 1 | + 0.745·3-s − 0.200·5-s − 1.39i·7-s − 0.444·9-s − 0.609·11-s − 0.384·13-s − 0.149·15-s − 1.28i·17-s − 0.706·19-s − 1.04i·21-s + 0.425i·23-s − 0.959·25-s − 1.07·27-s + (−0.655 + 0.755i)29-s + 1.51·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.155995339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155995339\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + (19 - 21.9i)T \) |
| good | 3 | \( 1 - 2.23T + 9T^{2} \) |
| 5 | \( 1 + T + 25T^{2} \) |
| 7 | \( 1 + 9.79iT - 49T^{2} \) |
| 11 | \( 1 + 6.70T + 121T^{2} \) |
| 13 | \( 1 + 5T + 169T^{2} \) |
| 17 | \( 1 + 21.9iT - 289T^{2} \) |
| 19 | \( 1 + 13.4T + 361T^{2} \) |
| 23 | \( 1 - 9.79iT - 529T^{2} \) |
| 31 | \( 1 - 46.9T + 961T^{2} \) |
| 37 | \( 1 + 43.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 65.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 20.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 48.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 39.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 97.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 87.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 21.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 65.7iT - 9.40e3T^{2} \) |
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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
−10.49156059722590037991833456866, −9.587969135992837255224623446592, −8.683698075599570073585841456290, −7.58751948716157053072682926234, −7.24988285806127796704024020748, −5.73544982696109542305316200624, −4.48798097603608245134869880194, −3.49797767753964001483475217326, −2.33425959852771949953199250012, −0.40023343899864517838490264464,
2.14470850582438567239106206626, 2.91341728651881546473606179830, 4.29717751501363784530427031168, 5.61263173017018409296044504506, 6.35088942601619210889828790583, 8.078769420845042877241601825855, 8.212334584306326314916249840380, 9.247967832994511810656937441430, 10.10463655194870474895529871694, 11.26249538700673638014257273844
