| L(s) = 1 | + 2.54·3-s − 0.818·5-s − 2.13·7-s + 3.49·9-s + 0.530·11-s + 1.02·13-s − 2.08·15-s − 7.45·17-s + 0.396·19-s − 5.43·21-s − 4.32·25-s + 1.26·27-s − 8.23·29-s + 7.63·31-s + 1.35·33-s + 1.74·35-s − 1.74·37-s + 2.60·39-s + 8.78·41-s − 10.6·43-s − 2.86·45-s − 4.39·47-s − 2.45·49-s − 18.9·51-s + 5.42·53-s − 0.433·55-s + 1.00·57-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.366·5-s − 0.806·7-s + 1.16·9-s + 0.159·11-s + 0.283·13-s − 0.538·15-s − 1.80·17-s + 0.0908·19-s − 1.18·21-s − 0.865·25-s + 0.242·27-s − 1.52·29-s + 1.37·31-s + 0.235·33-s + 0.295·35-s − 0.287·37-s + 0.416·39-s + 1.37·41-s − 1.61·43-s − 0.426·45-s − 0.641·47-s − 0.350·49-s − 2.65·51-s + 0.745·53-s − 0.0585·55-s + 0.133·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 5 | \( 1 + 0.818T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 0.530T + 11T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 + 7.45T + 17T^{2} \) |
| 19 | \( 1 - 0.396T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 7.63T + 31T^{2} \) |
| 37 | \( 1 + 1.74T + 37T^{2} \) |
| 41 | \( 1 - 8.78T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 4.39T + 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 + 7.80T + 59T^{2} \) |
| 61 | \( 1 + 7.82T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 9.63T + 83T^{2} \) |
| 89 | \( 1 - 0.481T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−8.054353018870004853865434967786, −7.52324912288524960750856601572, −6.64697704668670673002956005242, −6.07059492900653411854065656650, −4.75264421269348729504611231220, −3.97016098222876173617548951766, −3.37778308152555629243357025272, −2.56020555137184063558917412267, −1.74139777217839552616782994163, 0,
1.74139777217839552616782994163, 2.56020555137184063558917412267, 3.37778308152555629243357025272, 3.97016098222876173617548951766, 4.75264421269348729504611231220, 6.07059492900653411854065656650, 6.64697704668670673002956005242, 7.52324912288524960750856601572, 8.054353018870004853865434967786
