| L(s) = 1 | + 1.78·5-s + 4.26i·7-s + 1.97i·11-s − 2.73i·13-s + 0.947·17-s + (4.04 + 1.63i)19-s + 8.44i·23-s − 1.82·25-s − 6.92i·29-s − 2.57·31-s + 7.60i·35-s + 9.36i·37-s + 3.11i·41-s − 5.13i·43-s − 1.05i·47-s + ⋯ |
| L(s) = 1 | + 0.796·5-s + 1.61i·7-s + 0.595i·11-s − 0.758i·13-s + 0.229·17-s + (0.927 + 0.374i)19-s + 1.76i·23-s − 0.364·25-s − 1.28i·29-s − 0.463·31-s + 1.28i·35-s + 1.53i·37-s + 0.485i·41-s − 0.782i·43-s − 0.154i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994730023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994730023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-4.04 - 1.63i)T \) |
| good | 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 4.26iT - 7T^{2} \) |
| 11 | \( 1 - 1.97iT - 11T^{2} \) |
| 13 | \( 1 + 2.73iT - 13T^{2} \) |
| 17 | \( 1 - 0.947T + 17T^{2} \) |
| 23 | \( 1 - 8.44iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 - 9.36iT - 37T^{2} \) |
| 41 | \( 1 - 3.11iT - 41T^{2} \) |
| 43 | \( 1 + 5.13iT - 43T^{2} \) |
| 47 | \( 1 + 1.05iT - 47T^{2} \) |
| 53 | \( 1 - 4.33iT - 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 61 | \( 1 - 2.43T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 - 9.48T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 + 3.91T + 79T^{2} \) |
| 83 | \( 1 - 1.85iT - 83T^{2} \) |
| 89 | \( 1 - 13.5iT - 89T^{2} \) |
| 97 | \( 1 + 8.99iT - 97T^{2} \) |
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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
−8.284138438682898976195021277676, −7.84735353614912141785097916378, −6.90483901423044597570406180332, −5.98740459825293454480093969024, −5.50101518015403974295488129259, −5.14934504408336413867885565920, −3.83785870125145103971544296457, −2.92704166413566948645068375131, −2.20571713351599699773148376776, −1.37384658272966188171353919313,
0.51739047651290185873974025302, 1.44516787653012208470791858979, 2.49565590919337657900915772182, 3.54966412924143493009648147284, 4.17132152755130109717507450308, 5.04394321463251754149207707954, 5.77880865460814099058011238704, 6.73529986294934882669632586852, 7.02297225109404882075312811388, 7.87698336319567313641633322699
