| L(s) = 1 | + 1.63·3-s − 0.530·5-s + 0.325·7-s − 0.324·9-s − 3.05·11-s + 1.58·13-s − 0.867·15-s − 2.75·17-s + 1.04·19-s + 0.532·21-s − 2.36·23-s − 4.71·25-s − 5.43·27-s − 6.32·29-s − 7.66·31-s − 4.99·33-s − 0.172·35-s − 0.416·37-s + 2.59·39-s + 5.37·41-s + 9.93·43-s + 0.171·45-s + 3.44·47-s − 6.89·49-s − 4.50·51-s − 1.87·53-s + 1.62·55-s + ⋯ |
| L(s) = 1 | + 0.944·3-s − 0.237·5-s + 0.122·7-s − 0.108·9-s − 0.921·11-s + 0.439·13-s − 0.223·15-s − 0.667·17-s + 0.238·19-s + 0.116·21-s − 0.493·23-s − 0.943·25-s − 1.04·27-s − 1.17·29-s − 1.37·31-s − 0.870·33-s − 0.0291·35-s − 0.0684·37-s + 0.415·39-s + 0.840·41-s + 1.51·43-s + 0.0256·45-s + 0.502·47-s − 0.984·49-s − 0.630·51-s − 0.258·53-s + 0.218·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 1.63T + 3T^{2} \) |
| 5 | \( 1 + 0.530T + 5T^{2} \) |
| 7 | \( 1 - 0.325T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 + 2.36T + 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 + 7.66T + 31T^{2} \) |
| 37 | \( 1 + 0.416T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 - 3.44T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 7.62T + 73T^{2} \) |
| 79 | \( 1 + 8.44T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 5.95T + 89T^{2} \) |
| 97 | \( 1 - 4.93T + 97T^{2} \) |
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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
−8.434651968317793182504380006876, −7.73953204805443913365455405328, −7.32721348448823096741900786614, −6.02258365145246178109641185180, −5.46172691050048160108416818095, −4.26658080523793402206771984812, −3.57805237555831690677992792162, −2.63144584126770712915903476409, −1.83277875538880736394891838034, 0,
1.83277875538880736394891838034, 2.63144584126770712915903476409, 3.57805237555831690677992792162, 4.26658080523793402206771984812, 5.46172691050048160108416818095, 6.02258365145246178109641185180, 7.32721348448823096741900786614, 7.73953204805443913365455405328, 8.434651968317793182504380006876
