| L(s) = 1 | + 0.799·3-s + 0.626·5-s − 0.555·7-s − 2.36·9-s − 1.34·11-s + 2.90·13-s + 0.500·15-s + 2.10·17-s − 8.54·19-s − 0.444·21-s + 8.01·23-s − 4.60·25-s − 4.28·27-s − 2.10·29-s + 3.16·31-s − 1.07·33-s − 0.347·35-s − 10.9·37-s + 2.31·39-s − 0.458·41-s + 2.58·43-s − 1.47·45-s + 10.5·47-s − 6.69·49-s + 1.68·51-s − 5.54·53-s − 0.841·55-s + ⋯ |
| L(s) = 1 | + 0.461·3-s + 0.279·5-s − 0.209·7-s − 0.787·9-s − 0.405·11-s + 0.804·13-s + 0.129·15-s + 0.510·17-s − 1.96·19-s − 0.0969·21-s + 1.67·23-s − 0.921·25-s − 0.824·27-s − 0.390·29-s + 0.568·31-s − 0.186·33-s − 0.0587·35-s − 1.80·37-s + 0.371·39-s − 0.0716·41-s + 0.394·43-s − 0.220·45-s + 1.53·47-s − 0.955·49-s + 0.235·51-s − 0.761·53-s − 0.113·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.799T + 3T^{2} \) |
| 5 | \( 1 - 0.626T + 5T^{2} \) |
| 7 | \( 1 + 0.555T + 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 + 8.54T + 19T^{2} \) |
| 23 | \( 1 - 8.01T + 23T^{2} \) |
| 29 | \( 1 + 2.10T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 0.458T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 5.54T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 0.357T + 61T^{2} \) |
| 67 | \( 1 + 2.78T + 67T^{2} \) |
| 71 | \( 1 - 4.26T + 71T^{2} \) |
| 73 | \( 1 + 4.79T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−8.242195246903637049292342207036, −7.43372384202662911834825850551, −6.51661994202156741414214953924, −5.91109562931530586530370951597, −5.16754964630412924715506452546, −4.14819118318136014142513511425, −3.29840119231638671767517140900, −2.56503321835555810977737955729, −1.56585827858552894350308590414, 0,
1.56585827858552894350308590414, 2.56503321835555810977737955729, 3.29840119231638671767517140900, 4.14819118318136014142513511425, 5.16754964630412924715506452546, 5.91109562931530586530370951597, 6.51661994202156741414214953924, 7.43372384202662911834825850551, 8.242195246903637049292342207036
