| L(s) = 1 | + 2-s + 0.558·3-s + 4-s − 0.515·5-s + 0.558·6-s − 5.08·7-s + 8-s − 2.68·9-s − 0.515·10-s + 11-s + 0.558·12-s − 0.295·13-s − 5.08·14-s − 0.287·15-s + 16-s − 2.44·17-s − 2.68·18-s + 7.53·19-s − 0.515·20-s − 2.84·21-s + 22-s − 6.65·23-s + 0.558·24-s − 4.73·25-s − 0.295·26-s − 3.17·27-s − 5.08·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.322·3-s + 0.5·4-s − 0.230·5-s + 0.227·6-s − 1.92·7-s + 0.353·8-s − 0.896·9-s − 0.162·10-s + 0.301·11-s + 0.161·12-s − 0.0819·13-s − 1.35·14-s − 0.0742·15-s + 0.250·16-s − 0.593·17-s − 0.633·18-s + 1.72·19-s − 0.115·20-s − 0.619·21-s + 0.213·22-s − 1.38·23-s + 0.113·24-s − 0.946·25-s − 0.0579·26-s − 0.611·27-s − 0.961·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.172562477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172562477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.558T + 3T^{2} \) |
| 5 | \( 1 + 0.515T + 5T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 13 | \( 1 + 0.295T + 13T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 - 7.53T + 19T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 - 8.90T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 - 9.87T + 43T^{2} \) |
| 47 | \( 1 - 0.300T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 - 0.572T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 - 4.49T + 79T^{2} \) |
| 83 | \( 1 + 0.918T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.299655960611162702978119551291, −7.56362381494446191370119946432, −6.72817263895932367674149911832, −6.08045153737483048627287042575, −5.67748904138238103029640598286, −4.40609671530522060869444808450, −3.74075053954094243596755999238, −2.92774727124543572715989320353, −2.50957783615462081640538183795, −0.70219623617297957784873412618,
0.70219623617297957784873412618, 2.50957783615462081640538183795, 2.92774727124543572715989320353, 3.74075053954094243596755999238, 4.40609671530522060869444808450, 5.67748904138238103029640598286, 6.08045153737483048627287042575, 6.72817263895932367674149911832, 7.56362381494446191370119946432, 8.299655960611162702978119551291
