| L(s) = 1 | − 2.56·5-s − 3.12·7-s + 6.56·11-s − 0.561·13-s + 17-s − 7.68·19-s − 3.43·23-s + 1.56·25-s + 1.12·29-s + 8.24·31-s + 8·35-s + 4·37-s − 9.68·41-s + 7.68·43-s − 9.12·47-s + 2.75·49-s + 6·53-s − 16.8·55-s − 11.3·59-s + 4·61-s + 1.43·65-s + 12·67-s − 13.3·71-s − 8.24·73-s − 20.4·77-s − 2·79-s + 1.12·83-s + ⋯ |
| L(s) = 1 | − 1.14·5-s − 1.18·7-s + 1.97·11-s − 0.155·13-s + 0.242·17-s − 1.76·19-s − 0.716·23-s + 0.312·25-s + 0.208·29-s + 1.48·31-s + 1.35·35-s + 0.657·37-s − 1.51·41-s + 1.17·43-s − 1.33·47-s + 0.393·49-s + 0.824·53-s − 2.26·55-s − 1.48·59-s + 0.512·61-s + 0.178·65-s + 1.46·67-s − 1.58·71-s − 0.965·73-s − 2.33·77-s − 0.225·79-s + 0.123·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9777828110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9777828110\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 6.56T + 11T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 + 0.876T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−7.66659677771050967177732993215, −6.76079438504836472175884830853, −6.50325296523880377379050360535, −5.91141357152706708456380337190, −4.55033627638602931268610997158, −4.13282212674477747586179169477, −3.58212131614107307888554656106, −2.79346251848023181384386736454, −1.62997722817210187590595723170, −0.47169929649409843545845696047,
0.47169929649409843545845696047, 1.62997722817210187590595723170, 2.79346251848023181384386736454, 3.58212131614107307888554656106, 4.13282212674477747586179169477, 4.55033627638602931268610997158, 5.91141357152706708456380337190, 6.50325296523880377379050360535, 6.76079438504836472175884830853, 7.66659677771050967177732993215
