| L(s) = 1 | + 1.59·3-s − 1.91i·5-s − 0.700·7-s − 0.456·9-s + (−2.11 − 2.55i)11-s − 2.15·13-s − 3.05i·15-s + 1.75i·17-s + 5.53i·19-s − 1.11·21-s − 1.77i·23-s + 1.33·25-s − 5.51·27-s − 6.02·29-s − 0.652i·31-s + ⋯ |
| L(s) = 1 | + 0.920·3-s − 0.856i·5-s − 0.264·7-s − 0.152·9-s + (−0.638 − 0.769i)11-s − 0.599·13-s − 0.788i·15-s + 0.426i·17-s + 1.26i·19-s − 0.243·21-s − 0.370i·23-s + 0.266·25-s − 1.06·27-s − 1.11·29-s − 0.117i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4473158570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4473158570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (2.11 + 2.55i)T \) |
| good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 + 1.91iT - 5T^{2} \) |
| 7 | \( 1 + 0.700T + 7T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 - 1.75iT - 17T^{2} \) |
| 19 | \( 1 - 5.53iT - 19T^{2} \) |
| 23 | \( 1 + 1.77iT - 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 + 0.652iT - 31T^{2} \) |
| 37 | \( 1 + 9.42iT - 37T^{2} \) |
| 41 | \( 1 + 0.841iT - 41T^{2} \) |
| 43 | \( 1 - 4.78iT - 43T^{2} \) |
| 47 | \( 1 + 3.31iT - 47T^{2} \) |
| 53 | \( 1 + 2.59iT - 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 2.51T + 67T^{2} \) |
| 71 | \( 1 - 0.966iT - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 9.89T + 79T^{2} \) |
| 83 | \( 1 - 2.84iT - 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.207323486027989777644561364132, −8.089477084718416149833213430958, −7.07919274100914916718764375558, −5.86344340040056996334998971200, −5.45034980270043723861747565295, −4.31490075514612219835601860871, −3.49855698473731168684382671515, −2.66720867404822744116601911001, −1.64514974150136294533236784013, −0.11246675838428921639750065240,
1.91753207426450369096957999906, 2.87387283116816185207084265845, 3.14238630494337619074045197598, 4.46053593823406618188784697367, 5.22689479456342302485758569968, 6.30522216335569305339645291229, 7.17291963523047799222771308945, 7.53830475541472672066677236196, 8.411188877203014061866964317021, 9.317854699109660169311274575699
