| L(s) = 1 | − 2.41·3-s + 5-s − 2.41·7-s + 2.82·9-s + 2·13-s − 2.41·15-s + 5.65·17-s − 4.82·19-s + 5.82·21-s − 3.65·23-s + 25-s + 0.414·27-s + 2·29-s + 5.65·31-s − 2.41·35-s − 4·37-s − 4.82·39-s − 9.48·41-s − 3.58·43-s + 2.82·45-s − 7.58·47-s − 1.17·49-s − 13.6·51-s − 7.65·53-s + 11.6·57-s + 11.3·59-s + 61-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.447·5-s − 0.912·7-s + 0.942·9-s + 0.554·13-s − 0.623·15-s + 1.37·17-s − 1.10·19-s + 1.27·21-s − 0.762·23-s + 0.200·25-s + 0.0797·27-s + 0.371·29-s + 1.01·31-s − 0.408·35-s − 0.657·37-s − 0.773·39-s − 1.48·41-s − 0.546·43-s + 0.421·45-s − 1.10·47-s − 0.167·49-s − 1.91·51-s − 1.05·53-s + 1.54·57-s + 1.47·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−6.96056480412414674889323104537, −6.35503705140176780594227722634, −6.22651709729512000016468343765, −5.30721348410075514974921456278, −4.87692963999750943163856434019, −3.79869626696907905087233393187, −3.17044978635934989500553666789, −2.00293719542701549633808443107, −0.993201180404461114608620942651, 0,
0.993201180404461114608620942651, 2.00293719542701549633808443107, 3.17044978635934989500553666789, 3.79869626696907905087233393187, 4.87692963999750943163856434019, 5.30721348410075514974921456278, 6.22651709729512000016468343765, 6.35503705140176780594227722634, 6.96056480412414674889323104537
