| L(s) = 1 | + 3-s + i·5-s + 3.56i·7-s + 9-s + 1.56i·11-s + (−0.561 − 3.56i)13-s + i·15-s + 0.438·17-s + 5.12i·19-s + 3.56i·21-s + 1.56·23-s − 25-s + 27-s − 2·29-s + 2i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447i·5-s + 1.34i·7-s + 0.333·9-s + 0.470i·11-s + (−0.155 − 0.987i)13-s + 0.258i·15-s + 0.106·17-s + 1.17i·19-s + 0.777i·21-s + 0.325·23-s − 0.200·25-s + 0.192·27-s − 0.371·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.823537717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.823537717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (0.561 + 3.56i)T \) |
| good | 7 | \( 1 - 3.56iT - 7T^{2} \) |
| 11 | \( 1 - 1.56iT - 11T^{2} \) |
| 17 | \( 1 - 0.438T + 17T^{2} \) |
| 19 | \( 1 - 5.12iT - 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 - 5.56iT - 37T^{2} \) |
| 41 | \( 1 - 6.68iT - 41T^{2} \) |
| 43 | \( 1 - 0.876T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 + 6.24iT - 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 - 15.3iT - 67T^{2} \) |
| 71 | \( 1 + 9.56iT - 71T^{2} \) |
| 73 | \( 1 - 0.876iT - 73T^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 - 3.12iT - 83T^{2} \) |
| 89 | \( 1 - 0.438iT - 89T^{2} \) |
| 97 | \( 1 - 1.56iT - 97T^{2} \) |
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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
−9.699929991186183363254118588917, −8.788613863846994526816513929325, −8.114776353564899500432182220599, −7.43621512714403794527015445611, −6.34472503543136095429871376847, −5.62568690436271476179538796313, −4.70572209915381456133256806865, −3.39819438303702252786460812972, −2.73534645922557208625237661024, −1.68717261065427266987828045144,
0.65794469129480809042476282555, 1.94587392207633936538331016138, 3.25388519388819585016307901428, 4.16286404355042877168338383969, 4.78368970292020282905180182126, 6.03098227173383583197813891799, 7.12151633480150168634057872694, 7.44477906900694500475519734528, 8.550089373838776699232118267137, 9.161924582195718882628434774511
