| L(s) = 1 | + (−4.52 − 2.12i)5-s + 0.343i·7-s + 3.68i·11-s + 5.50i·13-s − 6.36·17-s + 2.62·19-s + 12.2·23-s + (15.9 + 19.2i)25-s − 24.0i·29-s + 1.04·31-s + (0.729 − 1.55i)35-s + 36.2i·37-s + 8.06i·41-s − 19.3i·43-s − 38.7·47-s + ⋯ |
| L(s) = 1 | + (−0.905 − 0.425i)5-s + 0.0490i·7-s + 0.334i·11-s + 0.423i·13-s − 0.374·17-s + 0.138·19-s + 0.533·23-s + (0.638 + 0.769i)25-s − 0.828i·29-s + 0.0337·31-s + (0.0208 − 0.0443i)35-s + 0.979i·37-s + 0.196i·41-s − 0.449i·43-s − 0.824·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8023212371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8023212371\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.52 + 2.12i)T \) |
| good | 7 | \( 1 - 0.343iT - 49T^{2} \) |
| 11 | \( 1 - 3.68iT - 121T^{2} \) |
| 13 | \( 1 - 5.50iT - 169T^{2} \) |
| 17 | \( 1 + 6.36T + 289T^{2} \) |
| 19 | \( 1 - 2.62T + 361T^{2} \) |
| 23 | \( 1 - 12.2T + 529T^{2} \) |
| 29 | \( 1 + 24.0iT - 841T^{2} \) |
| 31 | \( 1 - 1.04T + 961T^{2} \) |
| 37 | \( 1 - 36.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.06iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 19.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 3.24T + 2.80e3T^{2} \) |
| 59 | \( 1 + 51.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 77.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 108. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 78.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 62.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 35.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 65.0iT - 9.40e3T^{2} \) |
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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
−8.588166082381677765579894842258, −7.932571825551234024555583618261, −7.13784800691936532197375504633, −6.43392517321128807777463054679, −5.29335655098220426069675061575, −4.55510904668067519021526723912, −3.82432628320574397991349799721, −2.77909068115124576250540886301, −1.51812651895400436823922631032, −0.23867971477353814936299796341,
0.997125841408104779133599476437, 2.52491530850969156188184224216, 3.37457636922120676459141814630, 4.17705946154059044563331703586, 5.12057140836465399590188353940, 6.03894087675389140713774893143, 7.00326143002622956414212829899, 7.48447825507426037543678231220, 8.412373331070365500022676185680, 8.938737787077677987459985893260
