| L(s) = 1 | − 0.887·2-s − 3.06·3-s − 1.21·4-s + 2.71·6-s + 3.56·7-s + 2.85·8-s + 6.37·9-s − 0.942·11-s + 3.70·12-s − 3.16·14-s − 0.109·16-s + 0.959·17-s − 5.66·18-s + 3.33·19-s − 10.9·21-s + 0.837·22-s − 6.59·23-s − 8.73·24-s − 10.3·27-s − 4.31·28-s + 8.39·29-s − 4.13·31-s − 5.60·32-s + 2.88·33-s − 0.852·34-s − 7.72·36-s − 7.97·37-s + ⋯ |
| L(s) = 1 | − 0.627·2-s − 1.76·3-s − 0.605·4-s + 1.11·6-s + 1.34·7-s + 1.00·8-s + 2.12·9-s − 0.284·11-s + 1.07·12-s − 0.845·14-s − 0.0273·16-s + 0.232·17-s − 1.33·18-s + 0.766·19-s − 2.38·21-s + 0.178·22-s − 1.37·23-s − 1.78·24-s − 1.98·27-s − 0.815·28-s + 1.55·29-s − 0.743·31-s − 0.991·32-s + 0.502·33-s − 0.146·34-s − 1.28·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.887T + 2T^{2} \) |
| 3 | \( 1 + 3.06T + 3T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 + 0.942T + 11T^{2} \) |
| 17 | \( 1 - 0.959T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 + 7.97T + 37T^{2} \) |
| 41 | \( 1 - 0.797T + 41T^{2} \) |
| 43 | \( 1 + 9.00T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 - 1.00T + 59T^{2} \) |
| 61 | \( 1 - 4.52T + 61T^{2} \) |
| 67 | \( 1 - 1.11T + 67T^{2} \) |
| 71 | \( 1 - 1.85T + 71T^{2} \) |
| 73 | \( 1 + 4.69T + 73T^{2} \) |
| 79 | \( 1 - 5.64T + 79T^{2} \) |
| 83 | \( 1 + 0.187T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.062629506437459503637534975426, −7.36680020649322921206991223107, −6.58426679975275340390326961748, −5.60103691583553887823901595127, −5.06899208883506724368512097599, −4.65112085327708918913505167261, −3.71139548520302061117333964025, −1.83974272021934189086141366491, −1.09728164405583043541236890144, 0,
1.09728164405583043541236890144, 1.83974272021934189086141366491, 3.71139548520302061117333964025, 4.65112085327708918913505167261, 5.06899208883506724368512097599, 5.60103691583553887823901595127, 6.58426679975275340390326961748, 7.36680020649322921206991223107, 8.062629506437459503637534975426
