| L(s) = 1 | + 3-s − 5-s − 1.73·7-s − 2·9-s − 3.46·13-s − 15-s + 6.92·17-s − 3.46·19-s − 1.73·21-s + 25-s − 5·27-s + 8·31-s + 1.73·35-s − 8·37-s − 3.46·39-s − 12.1·41-s + 8.66·43-s + 2·45-s − 9·47-s − 4·49-s + 6.92·51-s + 6·53-s − 3.46·57-s + 12·59-s − 8.66·61-s + 3.46·63-s + 3.46·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.654·7-s − 0.666·9-s − 0.960·13-s − 0.258·15-s + 1.68·17-s − 0.794·19-s − 0.377·21-s + 0.200·25-s − 0.962·27-s + 1.43·31-s + 0.292·35-s − 1.31·37-s − 0.554·39-s − 1.89·41-s + 1.32·43-s + 0.298·45-s − 1.31·47-s − 0.571·49-s + 0.970·51-s + 0.824·53-s − 0.458·57-s + 1.56·59-s − 1.10·61-s + 0.436·63-s + 0.429·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.374915985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374915985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.66T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 8.66T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.83709131533844452858268260654, −7.01722249286178293796906355378, −6.45491356668435727724764755866, −5.54993343380825695056482810201, −4.99575025013761972477568994345, −4.00802485643878739847160852654, −3.27821681040554321825457911013, −2.83157271802150047233727642340, −1.86511929096749150762795889032, −0.51907694998472670152303365101,
0.51907694998472670152303365101, 1.86511929096749150762795889032, 2.83157271802150047233727642340, 3.27821681040554321825457911013, 4.00802485643878739847160852654, 4.99575025013761972477568994345, 5.54993343380825695056482810201, 6.45491356668435727724764755866, 7.01722249286178293796906355378, 7.83709131533844452858268260654
