| L(s) = 1 | + 2.96·3-s + 0.675·5-s + 5.77·9-s + 2·11-s + 3.63·13-s + 2·15-s + 5.24·17-s + 5.92·19-s − 23-s − 4.54·25-s + 8.20·27-s + 2.77·29-s − 1.61·31-s + 5.92·33-s − 2·37-s + 10.7·39-s + 3.63·41-s − 4·43-s + 3.89·45-s − 7.53·47-s + 15.5·51-s − 6·53-s + 1.35·55-s + 17.5·57-s + 12.5·59-s − 9.82·61-s + 2.45·65-s + ⋯ |
| L(s) = 1 | + 1.70·3-s + 0.301·5-s + 1.92·9-s + 0.603·11-s + 1.00·13-s + 0.516·15-s + 1.27·17-s + 1.35·19-s − 0.208·23-s − 0.908·25-s + 1.58·27-s + 0.514·29-s − 0.289·31-s + 1.03·33-s − 0.328·37-s + 1.72·39-s + 0.568·41-s − 0.609·43-s + 0.581·45-s − 1.09·47-s + 2.17·51-s − 0.824·53-s + 0.182·55-s + 2.32·57-s + 1.63·59-s − 1.25·61-s + 0.304·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.408225784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.408225784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 - 0.675T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 9.82T + 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 - 2.28T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 + 6.59T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.86374589230299134889452304695, −7.32329878633012555687258397699, −6.45511177702785518134255911200, −5.73610870795134825025065905125, −4.85297880289333531563750424984, −3.79670865165471952652807038458, −3.48401829331019576894076127017, −2.77952782753262497728702146604, −1.73835363993043496837750457554, −1.17613133600147158339982984959,
1.17613133600147158339982984959, 1.73835363993043496837750457554, 2.77952782753262497728702146604, 3.48401829331019576894076127017, 3.79670865165471952652807038458, 4.85297880289333531563750424984, 5.73610870795134825025065905125, 6.45511177702785518134255911200, 7.32329878633012555687258397699, 7.86374589230299134889452304695
