| L(s) = 1 | + 2.44·2-s + 3.95·4-s + 0.591·7-s + 4.77·8-s + 4.26·11-s + 5.06·13-s + 1.44·14-s + 3.73·16-s + 5.39·17-s − 5.73·19-s + 10.4·22-s − 2.12·23-s + 12.3·26-s + 2.33·28-s + 9.48·29-s − 1.38·31-s − 0.421·32-s + 13.1·34-s − 11.3·37-s − 14.0·38-s − 0.403·41-s + 5.30·43-s + 16.8·44-s − 5.18·46-s − 8.60·47-s − 6.65·49-s + 20.0·52-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 1.97·4-s + 0.223·7-s + 1.68·8-s + 1.28·11-s + 1.40·13-s + 0.385·14-s + 0.934·16-s + 1.30·17-s − 1.31·19-s + 2.21·22-s − 0.443·23-s + 2.42·26-s + 0.441·28-s + 1.76·29-s − 0.249·31-s − 0.0745·32-s + 2.25·34-s − 1.86·37-s − 2.27·38-s − 0.0630·41-s + 0.808·43-s + 2.54·44-s − 0.764·46-s − 1.25·47-s − 0.950·49-s + 2.77·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.117420561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.117420561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 7 | \( 1 - 0.591T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 - 5.39T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 + 2.12T + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 0.403T + 41T^{2} \) |
| 43 | \( 1 - 5.30T + 43T^{2} \) |
| 47 | \( 1 + 8.60T + 47T^{2} \) |
| 53 | \( 1 + 0.337T + 53T^{2} \) |
| 59 | \( 1 + 2.73T + 59T^{2} \) |
| 61 | \( 1 - 9.01T + 61T^{2} \) |
| 67 | \( 1 + 5.86T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 - 8.44T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
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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
−8.136653229475508229163304499827, −6.95895636189998550636997785299, −6.44804759602033338546057883864, −5.99932662355692554834460724947, −5.15124253882070571815159940633, −4.42186287960711395877325973500, −3.68287144609597562002996531682, −3.30649128588012274368488243951, −2.06720629114833855366501895183, −1.23238426502359041709805812486,
1.23238426502359041709805812486, 2.06720629114833855366501895183, 3.30649128588012274368488243951, 3.68287144609597562002996531682, 4.42186287960711395877325973500, 5.15124253882070571815159940633, 5.99932662355692554834460724947, 6.44804759602033338546057883864, 6.95895636189998550636997785299, 8.136653229475508229163304499827
