| L(s) = 1 | + 1.15·3-s + 4.11·5-s − 1.30·7-s − 1.66·9-s + 11-s − 6.30·13-s + 4.74·15-s − 3.49·17-s − 7.02·19-s − 1.50·21-s − 4.30·23-s + 11.9·25-s − 5.38·27-s + 7.31·29-s − 4.67·31-s + 1.15·33-s − 5.38·35-s − 7.08·37-s − 7.26·39-s − 5.46·41-s − 0.276·43-s − 6.86·45-s + 2.61·47-s − 5.28·49-s − 4.02·51-s + 53-s + 4.11·55-s + ⋯ |
| L(s) = 1 | + 0.666·3-s + 1.84·5-s − 0.494·7-s − 0.556·9-s + 0.301·11-s − 1.74·13-s + 1.22·15-s − 0.846·17-s − 1.61·19-s − 0.329·21-s − 0.898·23-s + 2.38·25-s − 1.03·27-s + 1.35·29-s − 0.840·31-s + 0.200·33-s − 0.910·35-s − 1.16·37-s − 1.16·39-s − 0.852·41-s − 0.0421·43-s − 1.02·45-s + 0.381·47-s − 0.755·49-s − 0.564·51-s + 0.137·53-s + 0.554·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 13 | \( 1 + 6.30T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 + 7.02T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 - 7.31T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 + 7.08T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 0.276T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 - 6.90T + 61T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 + 5.94T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 + 5.08T + 97T^{2} \) |
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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
−8.205101379576255862540105621694, −6.90730222174325794577836986090, −6.62587825686808856796032263735, −5.76214212140930672110727637720, −5.11592732766052076088801197368, −4.20873481568555858687282881930, −3.00204138020397601731135265631, −2.29441873891986210534033046769, −1.90423509786434676964396975702, 0,
1.90423509786434676964396975702, 2.29441873891986210534033046769, 3.00204138020397601731135265631, 4.20873481568555858687282881930, 5.11592732766052076088801197368, 5.76214212140930672110727637720, 6.62587825686808856796032263735, 6.90730222174325794577836986090, 8.205101379576255862540105621694
