| L(s) = 1 | − 2.56·3-s + 5-s − 2.56·7-s + 3.56·9-s + 11-s + 2·13-s − 2.56·15-s − 0.561·17-s + 2.56·19-s + 6.56·21-s + 5.12·23-s + 25-s − 1.43·27-s + 9.68·29-s + 6.56·31-s − 2.56·33-s − 2.56·35-s − 5.68·37-s − 5.12·39-s + 2·41-s + 10.2·43-s + 3.56·45-s − 13.1·47-s − 0.438·49-s + 1.43·51-s + 4.56·53-s + 55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.447·5-s − 0.968·7-s + 1.18·9-s + 0.301·11-s + 0.554·13-s − 0.661·15-s − 0.136·17-s + 0.587·19-s + 1.43·21-s + 1.06·23-s + 0.200·25-s − 0.276·27-s + 1.79·29-s + 1.17·31-s − 0.445·33-s − 0.432·35-s − 0.934·37-s − 0.820·39-s + 0.312·41-s + 1.56·43-s + 0.530·45-s − 1.91·47-s − 0.0626·49-s + 0.201·51-s + 0.626·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8514647720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8514647720\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 19 | \( 1 - 2.56T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 9.68T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 4.56T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 - 2.31T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.12T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 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
−11.14126960236329538149896188774, −10.31578290271914889975618761475, −9.610102380261814035787365797523, −8.506330913515456389452192318983, −6.90714749289251182514887539182, −6.42240231639132883898098202212, −5.55670259443597983322528411319, −4.56056513454979795834442771016, −3.04747812934299332212872036183, −0.963725731828977312567788341922,
0.963725731828977312567788341922, 3.04747812934299332212872036183, 4.56056513454979795834442771016, 5.55670259443597983322528411319, 6.42240231639132883898098202212, 6.90714749289251182514887539182, 8.506330913515456389452192318983, 9.610102380261814035787365797523, 10.31578290271914889975618761475, 11.14126960236329538149896188774
