| L(s) = 1 | − 3.86·3-s + 4.67·5-s + 5.97·9-s + 14.5i·11-s + 12.7·13-s − 18.1·15-s + 19.5i·17-s − 17.7·19-s − 8.86·23-s − 3.11·25-s + 11.7·27-s − 35.4i·29-s + 29.0i·31-s − 56.4i·33-s + 12.2i·37-s + ⋯ |
| L(s) = 1 | − 1.28·3-s + 0.935·5-s + 0.663·9-s + 1.32i·11-s + 0.977·13-s − 1.20·15-s + 1.14i·17-s − 0.932·19-s − 0.385·23-s − 0.124·25-s + 0.433·27-s − 1.22i·29-s + 0.936i·31-s − 1.71i·33-s + 0.330i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6611356160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6611356160\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.86T + 9T^{2} \) |
| 5 | \( 1 - 4.67T + 25T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 12.7T + 169T^{2} \) |
| 17 | \( 1 - 19.5iT - 289T^{2} \) |
| 19 | \( 1 + 17.7T + 361T^{2} \) |
| 23 | \( 1 + 8.86T + 529T^{2} \) |
| 29 | \( 1 + 35.4iT - 841T^{2} \) |
| 31 | \( 1 - 29.0iT - 961T^{2} \) |
| 37 | \( 1 - 12.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 22.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 79.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 42.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 36.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 2.40T + 3.48e3T^{2} \) |
| 61 | \( 1 - 29.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 40.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 22.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 76.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 136.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 49.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 1.12iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 158. iT - 9.40e3T^{2} \) |
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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
−9.830215730018176259668201115672, −8.870167894045156440123942395307, −7.972482722392912169884873234238, −6.79821316059766934618692881453, −6.20747818056340432397277885018, −5.69891961422694547223898058194, −4.73166593963425447216637775549, −3.91088304837458125367513321048, −2.24240929626321801201756070792, −1.37116556673803216474236932234,
0.22678696855785470199614494733, 1.30879890718330725258948457978, 2.68272809152670246994258931713, 3.89557726257828635808536430567, 5.07837732549814974963793427402, 5.76376070997047051047074510881, 6.20032244588348173355041106642, 6.96243519385349345591496095484, 8.249996264269795675514735758475, 8.936354157756706725265457290390
