| L(s) = 1 | + 3-s + 5-s + 3.18·7-s + 9-s − 3.36·13-s + 15-s + 5.18·17-s + 3.18·21-s + 23-s + 25-s + 27-s − 1.18·29-s + 2.17·31-s + 3.18·35-s + 9.53·37-s − 3.36·39-s − 6.55·41-s − 1.01·43-s + 45-s + 6.37·47-s + 3.17·49-s + 5.18·51-s − 2.55·53-s + 8.55·59-s − 3.01·61-s + 3.18·63-s − 3.36·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.20·7-s + 0.333·9-s − 0.932·13-s + 0.258·15-s + 1.25·17-s + 0.696·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.220·29-s + 0.390·31-s + 0.539·35-s + 1.56·37-s − 0.538·39-s − 1.02·41-s − 0.155·43-s + 0.149·45-s + 0.930·47-s + 0.453·49-s + 0.726·51-s − 0.350·53-s + 1.11·59-s − 0.386·61-s + 0.401·63-s − 0.417·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.968065646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.968065646\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 6.55T + 41T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 - 8.55T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 + 7.53T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.37T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 + 9.91T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−8.754395399057819355516872551052, −7.962740691291711267614675053910, −7.57371601122615747973926052969, −6.63028149616968766806720444926, −5.54598254167902757241732005586, −4.96237404235056616861471969400, −4.10022832181401538336552718079, −2.97407601419974883919205807925, −2.12290330002362793953796179573, −1.13662971583271083591384289217,
1.13662971583271083591384289217, 2.12290330002362793953796179573, 2.97407601419974883919205807925, 4.10022832181401538336552718079, 4.96237404235056616861471969400, 5.54598254167902757241732005586, 6.63028149616968766806720444926, 7.57371601122615747973926052969, 7.962740691291711267614675053910, 8.754395399057819355516872551052
