| L(s) = 1 | − 2.94·3-s − 1.59·5-s − 7-s + 5.69·9-s − 11-s + 13-s + 4.71·15-s − 4.45·17-s + 4.27·19-s + 2.94·21-s − 5.94·23-s − 2.44·25-s − 7.94·27-s + 3.52·29-s − 1.68·31-s + 2.94·33-s + 1.59·35-s − 4.39·37-s − 2.94·39-s + 9.25·41-s + 6.11·43-s − 9.10·45-s + 7.54·47-s + 49-s + 13.1·51-s − 2.93·53-s + 1.59·55-s + ⋯ |
| L(s) = 1 | − 1.70·3-s − 0.715·5-s − 0.377·7-s + 1.89·9-s − 0.301·11-s + 0.277·13-s + 1.21·15-s − 1.08·17-s + 0.981·19-s + 0.643·21-s − 1.23·23-s − 0.488·25-s − 1.52·27-s + 0.654·29-s − 0.303·31-s + 0.513·33-s + 0.270·35-s − 0.722·37-s − 0.472·39-s + 1.44·41-s + 0.932·43-s − 1.35·45-s + 1.10·47-s + 0.142·49-s + 1.84·51-s − 0.402·53-s + 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.94T + 3T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 17 | \( 1 + 4.45T + 17T^{2} \) |
| 19 | \( 1 - 4.27T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 + 1.68T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 - 9.25T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 - 7.54T + 47T^{2} \) |
| 53 | \( 1 + 2.93T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 + 8.22T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 + 0.129T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−7.39502090804359271185893879015, −6.69316002622238007461468730381, −5.95798777445653448648389784155, −5.63300723549121687192234687784, −4.58643244711180485047215901435, −4.23394011609497378198256773319, −3.28060415216096047944961782485, −2.06148906938658328450940433553, −0.839142224914833621474890261342, 0,
0.839142224914833621474890261342, 2.06148906938658328450940433553, 3.28060415216096047944961782485, 4.23394011609497378198256773319, 4.58643244711180485047215901435, 5.63300723549121687192234687784, 5.95798777445653448648389784155, 6.69316002622238007461468730381, 7.39502090804359271185893879015
