| L(s) = 1 | − 1.03·5-s + 2.44·7-s + 5.46·11-s − 4.24·13-s − 3.46·17-s − 0.535·19-s + 2.82·23-s − 3.92·25-s − 5.93·29-s + 7.34·31-s − 2.53·35-s + 9.14·37-s − 11.4·41-s − 3.46·43-s − 2.82·47-s − 1.00·49-s − 9.52·53-s − 5.65·55-s − 13.8·59-s − 9.14·61-s + 4.39·65-s − 1.07·67-s + 16.2·71-s + 4·73-s + 13.3·77-s − 2.44·79-s − 1.46·83-s + ⋯ |
| L(s) = 1 | − 0.462·5-s + 0.925·7-s + 1.64·11-s − 1.17·13-s − 0.840·17-s − 0.122·19-s + 0.589·23-s − 0.785·25-s − 1.10·29-s + 1.31·31-s − 0.428·35-s + 1.50·37-s − 1.79·41-s − 0.528·43-s − 0.412·47-s − 0.142·49-s − 1.30·53-s − 0.762·55-s − 1.80·59-s − 1.17·61-s + 0.544·65-s − 0.130·67-s + 1.92·71-s + 0.468·73-s + 1.52·77-s − 0.275·79-s − 0.160·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.93T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 9.52T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−7.48722597593821731351139816032, −6.62741363825573882943218528847, −6.23900273146574154654465124178, −5.03984612445584751386675329635, −4.62205581899848941529757752759, −3.97429355907882679609985055539, −3.09889389179598106124132983286, −2.03925025985158664096289788396, −1.34502652570077379521728212303, 0,
1.34502652570077379521728212303, 2.03925025985158664096289788396, 3.09889389179598106124132983286, 3.97429355907882679609985055539, 4.62205581899848941529757752759, 5.03984612445584751386675329635, 6.23900273146574154654465124178, 6.62741363825573882943218528847, 7.48722597593821731351139816032
