| L(s) = 1 | + 3-s + 5-s + 1.75·7-s + 9-s − 2.83·11-s + 4.33·13-s + 15-s + 6.47·17-s + 2.33·19-s + 1.75·21-s − 2.17·23-s + 25-s + 27-s + 5.93·29-s − 10.2·31-s − 2.83·33-s + 1.75·35-s + 7.52·37-s + 4.33·39-s + 9.89·41-s − 10.9·43-s + 45-s − 7.41·47-s − 3.90·49-s + 6.47·51-s − 4.31·53-s − 2.83·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.665·7-s + 0.333·9-s − 0.855·11-s + 1.20·13-s + 0.258·15-s + 1.57·17-s + 0.536·19-s + 0.383·21-s − 0.453·23-s + 0.200·25-s + 0.192·27-s + 1.10·29-s − 1.84·31-s − 0.493·33-s + 0.297·35-s + 1.23·37-s + 0.694·39-s + 1.54·41-s − 1.66·43-s + 0.149·45-s − 1.08·47-s − 0.557·49-s + 0.907·51-s − 0.593·53-s − 0.382·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.128928909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.128928909\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 + 2.83T + 11T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 4.31T + 53T^{2} \) |
| 59 | \( 1 - 5.22T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 0.855T + 79T^{2} \) |
| 83 | \( 1 - 2.27T + 83T^{2} \) |
| 89 | \( 1 - 7.22T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.163176265041306577518322667291, −8.023077299989352306554600326562, −7.14192785675806595897382494395, −6.11027718314346962258202861043, −5.48841415388514048860977970567, −4.74541945512577201416066458751, −3.66522395301175037066704208289, −3.01146049298271198026223079526, −1.93332321155591937100575075267, −1.07069515272113199371367906169,
1.07069515272113199371367906169, 1.93332321155591937100575075267, 3.01146049298271198026223079526, 3.66522395301175037066704208289, 4.74541945512577201416066458751, 5.48841415388514048860977970567, 6.11027718314346962258202861043, 7.14192785675806595897382494395, 8.023077299989352306554600326562, 8.163176265041306577518322667291
