L(s) = 1 | − 0.801·3-s + 1.69·7-s − 2.35·9-s + 0.911·11-s − 1.55·13-s − 5.29·17-s + 19-s − 1.35·21-s + 4.24·23-s + 4.29·27-s + 5.00·29-s − 1.82·31-s − 0.731·33-s − 6.29·37-s + 1.24·39-s + 4.18·41-s + 7.31·43-s − 2.04·47-s − 4.13·49-s + 4.24·51-s + 2.70·53-s − 0.801·57-s − 9.87·59-s + 0.542·61-s − 3.98·63-s − 13.9·67-s − 3.40·69-s + ⋯ |
L(s) = 1 | − 0.462·3-s + 0.639·7-s − 0.785·9-s + 0.274·11-s − 0.431·13-s − 1.28·17-s + 0.229·19-s − 0.296·21-s + 0.885·23-s + 0.826·27-s + 0.930·29-s − 0.328·31-s − 0.127·33-s − 1.03·37-s + 0.199·39-s + 0.652·41-s + 1.11·43-s − 0.298·47-s − 0.591·49-s + 0.594·51-s + 0.371·53-s − 0.106·57-s − 1.28·59-s + 0.0695·61-s − 0.502·63-s − 1.71·67-s − 0.410·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 0.801T + 3T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 0.911T + 11T^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 + 6.29T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 + 2.04T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 - 0.542T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 2.91T + 89T^{2} \) |
| 97 | \( 1 + 1.55T + 97T^{2} \) |
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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
−7.53263564810710381778577625302, −6.71314180768639174777900977224, −6.22975453060353168147993980750, −5.27478353678068160627102618998, −4.87055392071616813395011469202, −4.07579865150227238079918520742, −3.01456189507633647254229079886, −2.28629400595512651982593412862, −1.19158975922590861684963161738, 0,
1.19158975922590861684963161738, 2.28629400595512651982593412862, 3.01456189507633647254229079886, 4.07579865150227238079918520742, 4.87055392071616813395011469202, 5.27478353678068160627102618998, 6.22975453060353168147993980750, 6.71314180768639174777900977224, 7.53263564810710381778577625302