L(s) = 1 | + 1.19·3-s − 2.67·5-s + 2.81·7-s − 1.57·9-s − 2.79·11-s − 1.71·13-s − 3.18·15-s − 5.86·17-s + 8.02·19-s + 3.36·21-s − 2.42·23-s + 2.14·25-s − 5.46·27-s − 7.60·29-s − 1.28·31-s − 3.33·33-s − 7.52·35-s − 1.73·37-s − 2.04·39-s − 10.5·41-s + 1.96·43-s + 4.21·45-s + 4.67·47-s + 0.933·49-s − 6.99·51-s + 6.68·53-s + 7.46·55-s + ⋯ |
L(s) = 1 | + 0.688·3-s − 1.19·5-s + 1.06·7-s − 0.525·9-s − 0.842·11-s − 0.475·13-s − 0.823·15-s − 1.42·17-s + 1.84·19-s + 0.733·21-s − 0.505·23-s + 0.428·25-s − 1.05·27-s − 1.41·29-s − 0.231·31-s − 0.580·33-s − 1.27·35-s − 0.285·37-s − 0.327·39-s − 1.64·41-s + 0.299·43-s + 0.628·45-s + 0.681·47-s + 0.133·49-s − 0.978·51-s + 0.917·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 + T \) |
good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 + 1.71T + 13T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 - 8.02T + 19T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 1.96T + 43T^{2} \) |
| 47 | \( 1 - 4.67T + 47T^{2} \) |
| 53 | \( 1 - 6.68T + 53T^{2} \) |
| 59 | \( 1 + 2.20T + 59T^{2} \) |
| 61 | \( 1 - 4.64T + 61T^{2} \) |
| 67 | \( 1 + 8.77T + 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 + 6.99T + 73T^{2} \) |
| 79 | \( 1 + 2.09T + 79T^{2} \) |
| 83 | \( 1 + 1.44T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
| 97 | \( 1 + 7.89T + 97T^{2} \) |
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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
−8.923693197358726695632970874720, −8.387395327216152428546603926824, −7.56036478780460216117910500503, −7.31318248441760806887821170801, −5.66211830393460440907832737886, −4.89718019416516067194255168083, −3.94346175796511121920785347505, −3.01562716823717189674867178226, −1.94643947096676450066607275729, 0,
1.94643947096676450066607275729, 3.01562716823717189674867178226, 3.94346175796511121920785347505, 4.89718019416516067194255168083, 5.66211830393460440907832737886, 7.31318248441760806887821170801, 7.56036478780460216117910500503, 8.387395327216152428546603926824, 8.923693197358726695632970874720