L(s) = 1 | + (−1.22 + 1.22i)3-s + (−4.54 + 4.54i)5-s − 1.15·7-s − 2.99i·9-s + (−6.42 − 6.42i)11-s + (−14.8 − 14.8i)13-s − 11.1i·15-s + 15.0·17-s + (9.44 − 9.44i)19-s + (1.41 − 1.41i)21-s + 31.6·23-s − 16.3i·25-s + (3.67 + 3.67i)27-s + (5.51 + 5.51i)29-s + 20.3i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.909 + 0.909i)5-s − 0.164·7-s − 0.333i·9-s + (−0.584 − 0.584i)11-s + (−1.14 − 1.14i)13-s − 0.742i·15-s + 0.887·17-s + (0.497 − 0.497i)19-s + (0.0671 − 0.0671i)21-s + 1.37·23-s − 0.653i·25-s + (0.136 + 0.136i)27-s + (0.190 + 0.190i)29-s + 0.656i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9918010140\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9918010140\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
good | 5 | \( 1 + (4.54 - 4.54i)T - 25iT^{2} \) |
| 7 | \( 1 + 1.15T + 49T^{2} \) |
| 11 | \( 1 + (6.42 + 6.42i)T + 121iT^{2} \) |
| 13 | \( 1 + (14.8 + 14.8i)T + 169iT^{2} \) |
| 17 | \( 1 - 15.0T + 289T^{2} \) |
| 19 | \( 1 + (-9.44 + 9.44i)T - 361iT^{2} \) |
| 23 | \( 1 - 31.6T + 529T^{2} \) |
| 29 | \( 1 + (-5.51 - 5.51i)T + 841iT^{2} \) |
| 31 | \( 1 - 20.3iT - 961T^{2} \) |
| 37 | \( 1 + (50.3 - 50.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 52.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-42.2 - 42.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 27.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-20.1 + 20.1i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-69.0 - 69.0i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (0.992 + 0.992i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-77.0 + 77.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 44.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 3.56iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-90.2 + 90.2i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 68.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 161.T + 9.40e3T^{2} \) |
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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
−10.42352013734029462602178594346, −9.462253453918568371679860376451, −8.284031556581103260892065678233, −7.46455032798956233495577605086, −6.82206317524476297527090226225, −5.51038616782717510729464036250, −4.86634285272079039157558756870, −3.32361728498179344191145298056, −2.97317661331514099642339588622, −0.57159451261384116810293567982,
0.75424077854937566497575714581, 2.21688943085660190732740324151, 3.76386408708631357780990352836, 4.82365293898242229833404553783, 5.40273317593187669791514615873, 6.86578024788488817371076267125, 7.46778525996510428846031773343, 8.246686271978905769141403525792, 9.285091576585217918228808555312, 10.00659254511575009121907711408