| L(s) = 1 | − 1.42·3-s − 5-s − 3.82·7-s − 0.967·9-s + 2.52·11-s + 7.13·13-s + 1.42·15-s − 6.37·17-s − 4.40·19-s + 5.45·21-s + 1.00·23-s + 25-s + 5.65·27-s + 2.93·29-s − 4.16·31-s − 3.59·33-s + 3.82·35-s − 11.7·37-s − 10.1·39-s + 1.67·41-s + 9.49·43-s + 0.967·45-s + 9.50·47-s + 7.64·49-s + 9.09·51-s + 5.55·53-s − 2.52·55-s + ⋯ |
| L(s) = 1 | − 0.823·3-s − 0.447·5-s − 1.44·7-s − 0.322·9-s + 0.760·11-s + 1.98·13-s + 0.368·15-s − 1.54·17-s − 1.01·19-s + 1.19·21-s + 0.210·23-s + 0.200·25-s + 1.08·27-s + 0.545·29-s − 0.747·31-s − 0.625·33-s + 0.646·35-s − 1.93·37-s − 1.62·39-s + 0.261·41-s + 1.44·43-s + 0.144·45-s + 1.38·47-s + 1.09·49-s + 1.27·51-s + 0.763·53-s − 0.340·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 1.42T + 3T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 - 7.13T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 + 4.40T + 19T^{2} \) |
| 23 | \( 1 - 1.00T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 1.67T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 - 9.50T + 47T^{2} \) |
| 53 | \( 1 - 5.55T + 53T^{2} \) |
| 59 | \( 1 - 5.03T + 59T^{2} \) |
| 61 | \( 1 + 6.50T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 4.97T + 73T^{2} \) |
| 79 | \( 1 + 5.11T + 79T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−7.10137916865734974044848963631, −6.70139782990497712926882896955, −6.10040541512899372208114795574, −5.74599585331704623470448774543, −4.53870635383769136518287107121, −3.87295936297229250312135033151, −3.33010614764775513617486291991, −2.25036510544533604110382119752, −0.932539278313875518478159747716, 0,
0.932539278313875518478159747716, 2.25036510544533604110382119752, 3.33010614764775513617486291991, 3.87295936297229250312135033151, 4.53870635383769136518287107121, 5.74599585331704623470448774543, 6.10040541512899372208114795574, 6.70139782990497712926882896955, 7.10137916865734974044848963631
