| L(s) = 1 | − 2-s + 0.542·3-s + 4-s + 0.203·5-s − 0.542·6-s − 2.28·7-s − 8-s − 2.70·9-s − 0.203·10-s + 0.221·11-s + 0.542·12-s + 2.44·13-s + 2.28·14-s + 0.110·15-s + 16-s + 1.65·17-s + 2.70·18-s + 1.02·19-s + 0.203·20-s − 1.23·21-s − 0.221·22-s + 23-s − 0.542·24-s − 4.95·25-s − 2.44·26-s − 3.09·27-s − 2.28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.312·3-s + 0.5·4-s + 0.0910·5-s − 0.221·6-s − 0.862·7-s − 0.353·8-s − 0.902·9-s − 0.0644·10-s + 0.0667·11-s + 0.156·12-s + 0.678·13-s + 0.609·14-s + 0.0285·15-s + 0.250·16-s + 0.401·17-s + 0.637·18-s + 0.234·19-s + 0.0455·20-s − 0.269·21-s − 0.0471·22-s + 0.208·23-s − 0.110·24-s − 0.991·25-s − 0.479·26-s − 0.595·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.542T + 3T^{2} \) |
| 5 | \( 1 - 0.203T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 - 0.221T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 - 1.02T + 19T^{2} \) |
| 31 | \( 1 + 0.610T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 + 9.63T + 41T^{2} \) |
| 43 | \( 1 + 8.26T + 43T^{2} \) |
| 47 | \( 1 + 0.772T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 1.20T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 - 8.99T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 4.54T + 97T^{2} \) |
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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
−9.286536246636758179528457532537, −8.416575399371136471362916587840, −7.86324192853829713966468455711, −6.72015912945576598217542263149, −6.12837648925918115314368737128, −5.17527565028895333180432040569, −3.59421489314926515280920663276, −3.00783367308752479142225379246, −1.68022475861967904016597614803, 0,
1.68022475861967904016597614803, 3.00783367308752479142225379246, 3.59421489314926515280920663276, 5.17527565028895333180432040569, 6.12837648925918115314368737128, 6.72015912945576598217542263149, 7.86324192853829713966468455711, 8.416575399371136471362916587840, 9.286536246636758179528457532537
