| L(s) = 1 | − 3-s + 3.34·5-s − 7-s + 9-s − 4.36·11-s + 1.50·13-s − 3.34·15-s + 5.56·17-s − 4.36·19-s + 21-s − 23-s + 6.17·25-s − 27-s − 5.73·29-s + 10.9·31-s + 4.36·33-s − 3.34·35-s − 0.336·37-s − 1.50·39-s + 7.82·41-s − 9.58·43-s + 3.34·45-s + 9.39·47-s + 49-s − 5.56·51-s + 1.31·53-s − 14.6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.49·5-s − 0.377·7-s + 0.333·9-s − 1.31·11-s + 0.418·13-s − 0.863·15-s + 1.34·17-s − 1.00·19-s + 0.218·21-s − 0.208·23-s + 1.23·25-s − 0.192·27-s − 1.06·29-s + 1.96·31-s + 0.760·33-s − 0.565·35-s − 0.0552·37-s − 0.241·39-s + 1.22·41-s − 1.46·43-s + 0.498·45-s + 1.36·47-s + 0.142·49-s − 0.779·51-s + 0.180·53-s − 1.96·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877537456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877537456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 29 | \( 1 + 5.73T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 0.336T + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 - 9.39T + 47T^{2} \) |
| 53 | \( 1 - 1.31T + 53T^{2} \) |
| 59 | \( 1 + 2.02T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 9.23T + 73T^{2} \) |
| 79 | \( 1 - 0.878T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−8.414005041590400892949822815576, −7.80649506368233894411151238164, −6.74958261861180496015021624833, −6.16060852505697532695142286289, −5.53487952530979549594983207824, −5.06091730135789754533156487357, −3.89003208791726907561584942359, −2.75894432622695631451642367556, −2.02927711936346008204352780018, −0.816634472394409674566550470649,
0.816634472394409674566550470649, 2.02927711936346008204352780018, 2.75894432622695631451642367556, 3.89003208791726907561584942359, 5.06091730135789754533156487357, 5.53487952530979549594983207824, 6.16060852505697532695142286289, 6.74958261861180496015021624833, 7.80649506368233894411151238164, 8.414005041590400892949822815576
