| L(s) = 1 | + 3-s + 5-s + 1.73·7-s − 2·9-s + 15-s − 6.92·19-s + 1.73·21-s + 25-s − 5·27-s + 2·31-s + 1.73·35-s − 8·37-s + 5.19·41-s + 5.19·43-s − 2·45-s + 3·47-s − 4·49-s − 12·53-s − 6.92·57-s + 6·59-s − 12.1·61-s − 3.46·63-s − 7·67-s − 6·71-s + 6.92·73-s + 75-s + 3.46·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.654·7-s − 0.666·9-s + 0.258·15-s − 1.58·19-s + 0.377·21-s + 0.200·25-s − 0.962·27-s + 0.359·31-s + 0.292·35-s − 1.31·37-s + 0.811·41-s + 0.792·43-s − 0.298·45-s + 0.437·47-s − 0.571·49-s − 1.64·53-s − 0.917·57-s + 0.781·59-s − 1.55·61-s − 0.436·63-s − 0.855·67-s − 0.712·71-s + 0.810·73-s + 0.115·75-s + 0.389·79-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−7.53425677247202445915154930283, −6.54483997956769996864119195578, −6.05756029874302393469627580298, −5.25930983228565534409246039580, −4.56002362830400627900733425879, −3.79852086146731192522019667812, −2.87653108983217856723446827956, −2.22544394005948352741040083431, −1.45890271118594015749355184423, 0,
1.45890271118594015749355184423, 2.22544394005948352741040083431, 2.87653108983217856723446827956, 3.79852086146731192522019667812, 4.56002362830400627900733425879, 5.25930983228565534409246039580, 6.05756029874302393469627580298, 6.54483997956769996864119195578, 7.53425677247202445915154930283
