| L(s) = 1 | − 2.82·3-s − 1.92·5-s − 7-s + 4.95·9-s + 11-s + 13-s + 5.41·15-s − 0.965·17-s + 3.09·19-s + 2.82·21-s − 3.72·23-s − 1.30·25-s − 5.51·27-s − 0.786·29-s + 2.78·31-s − 2.82·33-s + 1.92·35-s − 10.8·37-s − 2.82·39-s − 4.59·41-s − 4.94·43-s − 9.52·45-s − 0.799·47-s + 49-s + 2.72·51-s − 1.50·53-s − 1.92·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s − 0.859·5-s − 0.377·7-s + 1.65·9-s + 0.301·11-s + 0.277·13-s + 1.39·15-s − 0.234·17-s + 0.710·19-s + 0.615·21-s − 0.776·23-s − 0.261·25-s − 1.06·27-s − 0.146·29-s + 0.500·31-s − 0.491·33-s + 0.324·35-s − 1.77·37-s − 0.451·39-s − 0.716·41-s − 0.754·43-s − 1.41·45-s − 0.116·47-s + 0.142·49-s + 0.381·51-s − 0.206·53-s − 0.259·55-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 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 17 | \( 1 + 0.965T + 17T^{2} \) |
| 19 | \( 1 - 3.09T + 19T^{2} \) |
| 23 | \( 1 + 3.72T + 23T^{2} \) |
| 29 | \( 1 + 0.786T + 29T^{2} \) |
| 31 | \( 1 - 2.78T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 4.59T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 0.799T + 47T^{2} \) |
| 53 | \( 1 + 1.50T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 + 1.16T + 73T^{2} \) |
| 79 | \( 1 - 9.23T + 79T^{2} \) |
| 83 | \( 1 - 6.28T + 83T^{2} \) |
| 89 | \( 1 - 4.46T + 89T^{2} \) |
| 97 | \( 1 - 7.50T + 97T^{2} \) |
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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
−7.25097818306128926618823476249, −6.70064708272405805283117354958, −6.17449568453184682785326316200, −5.33288313249544019313626167610, −4.88737849779356985617641973236, −3.89258061732563178006546738317, −3.48220380921173053325003659774, −2.01025317676713245820816027077, −0.872745076257913355056231590505, 0,
0.872745076257913355056231590505, 2.01025317676713245820816027077, 3.48220380921173053325003659774, 3.89258061732563178006546738317, 4.88737849779356985617641973236, 5.33288313249544019313626167610, 6.17449568453184682785326316200, 6.70064708272405805283117354958, 7.25097818306128926618823476249
