| L(s) = 1 | − 2.73·3-s − 0.732·7-s + 4.46·9-s + 2·11-s − 3.46·13-s − 3.46·17-s + 0.535·19-s + 2·21-s + 6.19·23-s − 3.99·27-s − 6.92·29-s + 5.46·31-s − 5.46·33-s + 2·37-s + 9.46·39-s − 1.46·41-s − 5.26·43-s + 3.26·47-s − 6.46·49-s + 9.46·51-s − 11.4·53-s − 1.46·57-s + 7.46·59-s + 8.92·61-s − 3.26·63-s − 10.7·67-s − 16.9·69-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.276·7-s + 1.48·9-s + 0.603·11-s − 0.960·13-s − 0.840·17-s + 0.122·19-s + 0.436·21-s + 1.29·23-s − 0.769·27-s − 1.28·29-s + 0.981·31-s − 0.951·33-s + 0.328·37-s + 1.51·39-s − 0.228·41-s − 0.803·43-s + 0.476·47-s − 0.923·49-s + 1.32·51-s − 1.57·53-s − 0.193·57-s + 0.971·59-s + 1.14·61-s − 0.411·63-s − 1.31·67-s − 2.03·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7081207075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7081207075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.535T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 7.46T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.83650595303285476820263580208, −6.95075868802806106309828585066, −6.66837960765902862113328558396, −5.93451031960021610921731353165, −5.10685362924328498353103324138, −4.72182046056731014547531380533, −3.81379582276911341462078375010, −2.72976376768420681829077503874, −1.56998863245877578041370694572, −0.48503061838211880509911197607,
0.48503061838211880509911197607, 1.56998863245877578041370694572, 2.72976376768420681829077503874, 3.81379582276911341462078375010, 4.72182046056731014547531380533, 5.10685362924328498353103324138, 5.93451031960021610921731353165, 6.66837960765902862113328558396, 6.95075868802806106309828585066, 7.83650595303285476820263580208
