| L(s) = 1 | + 0.472·2-s − 2.02·3-s − 1.77·4-s − 5-s − 0.956·6-s − 4.07·7-s − 1.78·8-s + 1.10·9-s − 0.472·10-s − 11-s + 3.60·12-s − 4.33·13-s − 1.92·14-s + 2.02·15-s + 2.71·16-s + 3.91·17-s + 0.522·18-s + 3.80·19-s + 1.77·20-s + 8.24·21-s − 0.472·22-s + 5.26·23-s + 3.61·24-s + 25-s − 2.04·26-s + 3.83·27-s + 7.23·28-s + ⋯ |
| L(s) = 1 | + 0.333·2-s − 1.16·3-s − 0.888·4-s − 0.447·5-s − 0.390·6-s − 1.53·7-s − 0.630·8-s + 0.368·9-s − 0.149·10-s − 0.301·11-s + 1.03·12-s − 1.20·13-s − 0.513·14-s + 0.523·15-s + 0.678·16-s + 0.950·17-s + 0.123·18-s + 0.872·19-s + 0.397·20-s + 1.80·21-s − 0.100·22-s + 1.09·23-s + 0.737·24-s + 0.200·25-s − 0.401·26-s + 0.738·27-s + 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.472T + 2T^{2} \) |
| 3 | \( 1 + 2.02T + 3T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 - 5.26T + 23T^{2} \) |
| 29 | \( 1 - 8.29T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 + 8.57T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.63T + 59T^{2} \) |
| 61 | \( 1 + 4.25T + 61T^{2} \) |
| 67 | \( 1 + 0.607T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 4.38T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.074851936805631143944874076610, −7.00693768908510864899831888790, −6.65246334966723713346915670207, −5.51405931138271369834700439672, −5.28011703174695469290599959493, −4.48192943472933102772369691647, −3.33183556549461959545089176642, −2.96754332876516982140662704343, −0.832996661317017944786080942283, 0,
0.832996661317017944786080942283, 2.96754332876516982140662704343, 3.33183556549461959545089176642, 4.48192943472933102772369691647, 5.28011703174695469290599959493, 5.51405931138271369834700439672, 6.65246334966723713346915670207, 7.00693768908510864899831888790, 8.074851936805631143944874076610
