| L(s) = 1 | + 0.618·3-s − 0.236·5-s − 7-s − 2.61·9-s + 0.381·11-s + 5.47·13-s − 0.145·15-s + 0.854·17-s − 19-s − 0.618·21-s − 3.76·23-s − 4.94·25-s − 3.47·27-s + 2.38·29-s + 2.85·31-s + 0.236·33-s + 0.236·35-s + 3.76·37-s + 3.38·39-s − 8.56·41-s + 4.47·43-s + 0.618·45-s − 1.47·47-s + 49-s + 0.527·51-s + 5.09·53-s − 0.0901·55-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 0.105·5-s − 0.377·7-s − 0.872·9-s + 0.115·11-s + 1.51·13-s − 0.0376·15-s + 0.207·17-s − 0.229·19-s − 0.134·21-s − 0.784·23-s − 0.988·25-s − 0.668·27-s + 0.442·29-s + 0.512·31-s + 0.0410·33-s + 0.0399·35-s + 0.618·37-s + 0.541·39-s − 1.33·41-s + 0.681·43-s + 0.0921·45-s − 0.214·47-s + 0.142·49-s + 0.0739·51-s + 0.699·53-s − 0.0121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 0.236T + 5T^{2} \) |
| 11 | \( 1 - 0.381T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 - 0.854T + 17T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 + 8.56T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 + 0.236T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 8.23T + 71T^{2} \) |
| 73 | \( 1 - 1.38T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
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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
−7.70358526713145983046760719080, −6.54768029547702317031826722873, −6.11923575547962801968589177325, −5.56140887531726518289229923486, −4.48485686165145681225894877907, −3.73220606066158280875821739110, −3.18214951895971567753260337071, −2.29171160371927182175955713651, −1.27938777628015291316395541648, 0,
1.27938777628015291316395541648, 2.29171160371927182175955713651, 3.18214951895971567753260337071, 3.73220606066158280875821739110, 4.48485686165145681225894877907, 5.56140887531726518289229923486, 6.11923575547962801968589177325, 6.54768029547702317031826722873, 7.70358526713145983046760719080
