| L(s) = 1 | − i·3-s + i·5-s − 7-s − 9-s + 2.61i·11-s − 5.44i·13-s + 15-s − 1.53·17-s + 6.73i·19-s + i·21-s + 5.53·23-s − 25-s + i·27-s − 4.52i·29-s − 10.3·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 0.377·7-s − 0.333·9-s + 0.787i·11-s − 1.50i·13-s + 0.258·15-s − 0.371·17-s + 1.54i·19-s + 0.218i·21-s + 1.15·23-s − 0.200·25-s + 0.192i·27-s − 0.840i·29-s − 1.85·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08951371095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08951371095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2.61iT - 11T^{2} \) |
| 13 | \( 1 + 5.44iT - 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 6.73iT - 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 + 4.52iT - 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 1.53iT - 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 5.69iT - 43T^{2} \) |
| 47 | \( 1 + 0.469T + 47T^{2} \) |
| 53 | \( 1 + 4.44iT - 53T^{2} \) |
| 59 | \( 1 + 5.06iT - 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 - 15.9iT - 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 15.1iT - 83T^{2} \) |
| 89 | \( 1 + 9.55T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−8.804613696635715831739963495820, −8.117306053396242546104436958170, −7.27720019922838896672391735236, −6.98108799638848950732053430780, −5.77249261955410096004209591709, −5.52425009321037963588725047683, −4.18274451145439277115152380508, −3.32320944162749924494946858634, −2.49548078417743850494031806805, −1.44438572691970620125403582182,
0.02653787854387144856175401841, 1.50468591487360165941101501593, 2.77193389969163692859065921742, 3.57532442591697605755119649493, 4.54346515418992400435225198180, 5.04140099210687203962751915903, 6.03264844494641636791053837386, 6.79824143184825213967269722300, 7.44109327855501224255143381248, 8.735742709385967301565575309618
