| L(s) = 1 | + 1.06·2-s + 0.382·3-s − 0.866·4-s − 5-s + 0.406·6-s + 7-s − 3.05·8-s − 2.85·9-s − 1.06·10-s + 0.351·11-s − 0.331·12-s − 1.26·13-s + 1.06·14-s − 0.382·15-s − 1.51·16-s − 4.54·17-s − 3.03·18-s − 3.54·19-s + 0.866·20-s + 0.382·21-s + 0.373·22-s + 23-s − 1.16·24-s + 25-s − 1.34·26-s − 2.23·27-s − 0.866·28-s + ⋯ |
| L(s) = 1 | + 0.752·2-s + 0.220·3-s − 0.433·4-s − 0.447·5-s + 0.166·6-s + 0.377·7-s − 1.07·8-s − 0.951·9-s − 0.336·10-s + 0.105·11-s − 0.0956·12-s − 0.349·13-s + 0.284·14-s − 0.0987·15-s − 0.378·16-s − 1.10·17-s − 0.716·18-s − 0.814·19-s + 0.193·20-s + 0.0834·21-s + 0.0796·22-s + 0.208·23-s − 0.238·24-s + 0.200·25-s − 0.263·26-s − 0.430·27-s − 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 3 | \( 1 - 0.382T + 3T^{2} \) |
| 11 | \( 1 - 0.351T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 - 1.85T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 - 2.84T + 43T^{2} \) |
| 47 | \( 1 + 0.960T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 - 3.30T + 59T^{2} \) |
| 61 | \( 1 - 4.14T + 61T^{2} \) |
| 67 | \( 1 - 8.46T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 + 2.45T + 73T^{2} \) |
| 79 | \( 1 + 3.96T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−9.700449662036108251762570648166, −8.779220349356540780384292146495, −8.380280468122673201229219788780, −7.18493279170304499904739691141, −6.11337437403506205751982593246, −5.21157190056511651185282685384, −4.35500195121840774328663228492, −3.47880574629524555003728781967, −2.31584744518121774584039600656, 0,
2.31584744518121774584039600656, 3.47880574629524555003728781967, 4.35500195121840774328663228492, 5.21157190056511651185282685384, 6.11337437403506205751982593246, 7.18493279170304499904739691141, 8.380280468122673201229219788780, 8.779220349356540780384292146495, 9.700449662036108251762570648166
