| L(s) = 1 | − 1.67·2-s − 3-s + 0.815·4-s − 2.94·5-s + 1.67·6-s + 1.98·8-s + 9-s + 4.94·10-s − 3.66·11-s − 0.815·12-s − 1.91·13-s + 2.94·15-s − 4.96·16-s − 3.02·17-s − 1.67·18-s − 2.16·19-s − 2.40·20-s + 6.14·22-s − 23-s − 1.98·24-s + 3.69·25-s + 3.21·26-s − 27-s − 0.852·29-s − 4.94·30-s − 2.02·31-s + 4.35·32-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 0.577·3-s + 0.407·4-s − 1.31·5-s + 0.684·6-s + 0.702·8-s + 0.333·9-s + 1.56·10-s − 1.10·11-s − 0.235·12-s − 0.531·13-s + 0.761·15-s − 1.24·16-s − 0.734·17-s − 0.395·18-s − 0.496·19-s − 0.537·20-s + 1.31·22-s − 0.208·23-s − 0.405·24-s + 0.738·25-s + 0.631·26-s − 0.192·27-s − 0.158·29-s − 0.903·30-s − 0.363·31-s + 0.769·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03653427836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03653427836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 5 | \( 1 + 2.94T + 5T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 + 1.91T + 13T^{2} \) |
| 17 | \( 1 + 3.02T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 29 | \( 1 + 0.852T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 + 5.43T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 2.17T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 4.40T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 7.01T + 97T^{2} \) |
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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
−8.437253653954447786891654577564, −7.988658916104857080888501520295, −7.33648039640754971934896900573, −6.77496589835546505279496881763, −5.57850521352091369886522324342, −4.65406716216155966408496275537, −4.15725790154716842088192421163, −2.88171937638207211444761573002, −1.66434057404441930492852025061, −0.13812870122584921680516186017,
0.13812870122584921680516186017, 1.66434057404441930492852025061, 2.88171937638207211444761573002, 4.15725790154716842088192421163, 4.65406716216155966408496275537, 5.57850521352091369886522324342, 6.77496589835546505279496881763, 7.33648039640754971934896900573, 7.988658916104857080888501520295, 8.437253653954447786891654577564
