| L(s) = 1 | + 2·7-s + 3·11-s − 4·13-s − 3·17-s + 5·19-s − 6·23-s − 2·31-s + 2·37-s + 3·41-s + 4·43-s − 12·47-s − 3·49-s − 6·53-s − 2·61-s + 13·67-s + 12·71-s − 11·73-s + 6·77-s + 10·79-s − 9·83-s − 15·89-s − 8·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.904·11-s − 1.10·13-s − 0.727·17-s + 1.14·19-s − 1.25·23-s − 0.359·31-s + 0.328·37-s + 0.468·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.256·61-s + 1.58·67-s + 1.42·71-s − 1.28·73-s + 0.683·77-s + 1.12·79-s − 0.987·83-s − 1.58·89-s − 0.838·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.35737567121105, −15.89302018782934, −15.20383887143394, −14.61200328997100, −14.18898248551903, −13.84728647824518, −12.93887971480907, −12.41412656044995, −11.82166963580645, −11.34311679991434, −10.91349366847558, −9.852837913090186, −9.668085503757546, −8.999575187927093, −8.107942156644863, −7.833464539541286, −6.999199944557667, −6.502301619212354, −5.651098254122876, −5.019916463284809, −4.393243280298669, −3.749099832976063, −2.802901763414515, −1.994917705413413, −1.264482066466594, 0,
1.264482066466594, 1.994917705413413, 2.802901763414515, 3.749099832976063, 4.393243280298669, 5.019916463284809, 5.651098254122876, 6.502301619212354, 6.999199944557667, 7.833464539541286, 8.107942156644863, 8.999575187927093, 9.668085503757546, 9.852837913090186, 10.91349366847558, 11.34311679991434, 11.82166963580645, 12.41412656044995, 12.93887971480907, 13.84728647824518, 14.18898248551903, 14.61200328997100, 15.20383887143394, 15.89302018782934, 16.35737567121105
