| L(s) = 1 | − 1.02·2-s − 3-s − 0.956·4-s − 0.892·5-s + 1.02·6-s + 0.482·7-s + 3.02·8-s + 9-s + 0.911·10-s + 11-s + 0.956·12-s − 0.493·14-s + 0.892·15-s − 1.17·16-s − 0.548·17-s − 1.02·18-s − 0.444·19-s + 0.853·20-s − 0.482·21-s − 1.02·22-s + 2.39·23-s − 3.02·24-s − 4.20·25-s − 27-s − 0.461·28-s − 3.53·29-s − 0.911·30-s + ⋯ |
| L(s) = 1 | − 0.722·2-s − 0.577·3-s − 0.478·4-s − 0.398·5-s + 0.417·6-s + 0.182·7-s + 1.06·8-s + 0.333·9-s + 0.288·10-s + 0.301·11-s + 0.276·12-s − 0.131·14-s + 0.230·15-s − 0.293·16-s − 0.133·17-s − 0.240·18-s − 0.101·19-s + 0.190·20-s − 0.105·21-s − 0.217·22-s + 0.499·23-s − 0.616·24-s − 0.840·25-s − 0.192·27-s − 0.0872·28-s − 0.657·29-s − 0.166·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.02T + 2T^{2} \) |
| 5 | \( 1 + 0.892T + 5T^{2} \) |
| 7 | \( 1 - 0.482T + 7T^{2} \) |
| 17 | \( 1 + 0.548T + 17T^{2} \) |
| 19 | \( 1 + 0.444T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 + 2.07T + 37T^{2} \) |
| 41 | \( 1 + 9.47T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 + 0.984T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 - 5.22T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 - 9.60T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.75832759826069106017202567553, −7.33277865564921687355227870501, −6.42848980942688527310926157009, −5.62540764904598382344376048727, −4.80336193028949732228717813772, −4.22620376332942505626408151008, −3.40617025586638802057563909910, −2.01964796750797986448851453854, −1.03827943853907992085811025109, 0,
1.03827943853907992085811025109, 2.01964796750797986448851453854, 3.40617025586638802057563909910, 4.22620376332942505626408151008, 4.80336193028949732228717813772, 5.62540764904598382344376048727, 6.42848980942688527310926157009, 7.33277865564921687355227870501, 7.75832759826069106017202567553
