| L(s) = 1 | + (0.186 + 2.99i)3-s − 4.10i·5-s + (−8.93 + 1.11i)9-s + (12.3 − 0.764i)15-s + 15.3i·23-s + 8.11·25-s + (−5.00 − 26.5i)27-s − 61.5·31-s + 47.8·37-s + (4.58 + 36.6i)45-s − 79.5i·47-s − 49·49-s − 79.5i·53-s − 117. i·59-s − 94.5·67-s + ⋯ |
| L(s) = 1 | + (0.0620 + 0.998i)3-s − 0.821i·5-s + (−0.992 + 0.123i)9-s + (0.820 − 0.0509i)15-s + 0.668i·23-s + 0.324·25-s + (−0.185 − 0.982i)27-s − 1.98·31-s + 1.29·37-s + (0.101 + 0.815i)45-s − 1.69i·47-s − 0.999·49-s − 1.50i·53-s − 1.99i·59-s − 1.41·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0620 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0620 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9618090132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9618090132\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.186 - 2.99i)T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 4.10iT - 25T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 15.3iT - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 61.5T + 961T^{2} \) |
| 37 | \( 1 - 47.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 79.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 79.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 117. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 94.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 158. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 98.9T + 9.40e3T^{2} \) |
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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
−9.284450103608690878291109383995, −8.493210769216368287174315089631, −7.77155972091837373295758846738, −6.59921024082434947448413633980, −5.48786666977930026289186224992, −5.01609770302280479203240722076, −4.03655540870961985493187522788, −3.23625898424772009746075603553, −1.85312827904204486116315052203, −0.27199362273635443159274996744,
1.23484645433274026420894810506, 2.45529434778763073066572154833, 3.16874709978317348678697907573, 4.42784063020240870836328131869, 5.72459547652429360851119413154, 6.30993924630886364170689496932, 7.23093055028621573359847741006, 7.64130810981282567344464639347, 8.681031425657313909524228623885, 9.366891207237400834633700020620
