| L(s) = 1 | − 1.61·3-s + 0.618·5-s − 1.61·7-s − 0.381·9-s + 5.85·13-s − 1.00·15-s + 1.85·17-s − 4.85·19-s + 2.61·21-s − 4·23-s − 4.61·25-s + 5.47·27-s − 7.32·29-s + 1.09·31-s − 1.00·35-s − 9.61·37-s − 9.47·39-s + 9.61·41-s − 1.52·43-s − 0.236·45-s − 10.5·47-s − 4.38·49-s − 3·51-s + 0.618·53-s + 7.85·57-s − 2.38·59-s + 1.85·61-s + ⋯ |
| L(s) = 1 | − 0.934·3-s + 0.276·5-s − 0.611·7-s − 0.127·9-s + 1.62·13-s − 0.258·15-s + 0.449·17-s − 1.11·19-s + 0.571·21-s − 0.834·23-s − 0.923·25-s + 1.05·27-s − 1.36·29-s + 0.195·31-s − 0.169·35-s − 1.58·37-s − 1.51·39-s + 1.50·41-s − 0.232·43-s − 0.0351·45-s − 1.54·47-s − 0.625·49-s − 0.420·51-s + 0.0848·53-s + 1.04·57-s − 0.310·59-s + 0.237·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 13 | \( 1 - 5.85T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 + 9.61T + 37T^{2} \) |
| 41 | \( 1 - 9.61T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 0.618T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 - 1.85T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 5.09T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 - 3.85T + 79T^{2} \) |
| 83 | \( 1 + 9.38T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.718155373631337909562928826012, −8.790340375267244736428367243560, −7.993669350098141213010103018783, −6.72500884991025223316708225768, −6.02617977199967583271809473827, −5.59912022531615973356578863948, −4.23320561779896701245064714031, −3.28203777656175467468818893269, −1.70035795728857034723097312919, 0,
1.70035795728857034723097312919, 3.28203777656175467468818893269, 4.23320561779896701245064714031, 5.59912022531615973356578863948, 6.02617977199967583271809473827, 6.72500884991025223316708225768, 7.993669350098141213010103018783, 8.790340375267244736428367243560, 9.718155373631337909562928826012
