| L(s) = 1 | + 2.13·2-s + 3-s + 2.53·4-s − 1.46·5-s + 2.13·6-s − 1.13·7-s + 1.14·8-s + 9-s − 3.13·10-s + 2.53·12-s − 13-s − 2.40·14-s − 1.46·15-s − 2.63·16-s − 6.63·17-s + 2.13·18-s + 2.88·19-s − 3.73·20-s − 1.13·21-s − 2.09·23-s + 1.14·24-s − 2.84·25-s − 2.13·26-s + 27-s − 2.87·28-s + 0.645·29-s − 3.13·30-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.577·3-s + 1.26·4-s − 0.657·5-s + 0.869·6-s − 0.427·7-s + 0.406·8-s + 0.333·9-s − 0.989·10-s + 0.732·12-s − 0.277·13-s − 0.643·14-s − 0.379·15-s − 0.657·16-s − 1.60·17-s + 0.502·18-s + 0.663·19-s − 0.834·20-s − 0.246·21-s − 0.437·23-s + 0.234·24-s − 0.568·25-s − 0.417·26-s + 0.192·27-s − 0.542·28-s + 0.119·29-s − 0.571·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 17 | \( 1 + 6.63T + 17T^{2} \) |
| 19 | \( 1 - 2.88T + 19T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 - 0.645T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 + 0.708T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 + 0.268T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 + 7.94T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 8.71T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 4.57T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.62319958929401702141729785171, −7.16547389681303107638660719331, −6.33185464341646628175076701626, −5.68199425412987312221979461459, −4.65551402830599510981129793265, −4.22412795289324703663082355654, −3.46483982700736210047938094232, −2.78549769741706079903497932930, −1.90820538276815882913005718949, 0,
1.90820538276815882913005718949, 2.78549769741706079903497932930, 3.46483982700736210047938094232, 4.22412795289324703663082355654, 4.65551402830599510981129793265, 5.68199425412987312221979461459, 6.33185464341646628175076701626, 7.16547389681303107638660719331, 7.62319958929401702141729785171
