| L(s) = 1 | − 0.274·2-s + 3.09·3-s − 1.92·4-s − 5-s − 0.849·6-s + 3.63·7-s + 1.07·8-s + 6.57·9-s + 0.274·10-s + 2.45·11-s − 5.95·12-s + 3.81·13-s − 0.999·14-s − 3.09·15-s + 3.55·16-s − 3.62·17-s − 1.80·18-s + 1.92·20-s + 11.2·21-s − 0.674·22-s − 9.19·23-s + 3.33·24-s + 25-s − 1.04·26-s + 11.0·27-s − 7.00·28-s − 1.81·29-s + ⋯ |
| L(s) = 1 | − 0.194·2-s + 1.78·3-s − 0.962·4-s − 0.447·5-s − 0.346·6-s + 1.37·7-s + 0.381·8-s + 2.19·9-s + 0.0868·10-s + 0.740·11-s − 1.71·12-s + 1.05·13-s − 0.267·14-s − 0.798·15-s + 0.888·16-s − 0.878·17-s − 0.425·18-s + 0.430·20-s + 2.45·21-s − 0.143·22-s − 1.91·23-s + 0.680·24-s + 0.200·25-s − 0.205·26-s + 2.12·27-s − 1.32·28-s − 0.337·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.836866595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.836866595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.274T + 2T^{2} \) |
| 3 | \( 1 - 3.09T + 3T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 - 2.45T + 11T^{2} \) |
| 13 | \( 1 - 3.81T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 23 | \( 1 + 9.19T + 23T^{2} \) |
| 29 | \( 1 + 1.81T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 0.500T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 - 6.78T + 43T^{2} \) |
| 47 | \( 1 - 0.193T + 47T^{2} \) |
| 53 | \( 1 + 4.54T + 53T^{2} \) |
| 59 | \( 1 + 1.29T + 59T^{2} \) |
| 61 | \( 1 - 8.12T + 61T^{2} \) |
| 67 | \( 1 + 8.17T + 67T^{2} \) |
| 71 | \( 1 - 5.69T + 71T^{2} \) |
| 73 | \( 1 - 2.63T + 73T^{2} \) |
| 79 | \( 1 + 5.11T + 79T^{2} \) |
| 83 | \( 1 + 0.287T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−8.979661523405417431004886386867, −8.422714141469452362885988922092, −8.109792050270565138285193803752, −7.39894851181214725854035921747, −6.15008530166826905196912677705, −4.70532813259708660858285880400, −4.13839315757841603638506270687, −3.57996779403382287488232021506, −2.17431600312866013162514199720, −1.26202648240112478269414470827,
1.26202648240112478269414470827, 2.17431600312866013162514199720, 3.57996779403382287488232021506, 4.13839315757841603638506270687, 4.70532813259708660858285880400, 6.15008530166826905196912677705, 7.39894851181214725854035921747, 8.109792050270565138285193803752, 8.422714141469452362885988922092, 8.979661523405417431004886386867
