| L(s) = 1 | − 2·5-s + 2·7-s − 2·11-s − 13-s − 6·17-s + 6·19-s + 8·23-s − 25-s − 2·29-s − 10·31-s − 4·35-s − 6·37-s + 6·41-s − 4·43-s − 2·47-s − 3·49-s − 6·53-s + 4·55-s − 10·59-s − 2·61-s + 2·65-s − 10·67-s + 10·71-s + 2·73-s − 4·77-s + 4·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 0.603·11-s − 0.277·13-s − 1.45·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s − 1.79·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 0.291·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s − 1.30·59-s − 0.256·61-s + 0.248·65-s − 1.22·67-s + 1.18·71-s + 0.234·73-s − 0.455·77-s + 0.450·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.903536245670668621130507435375, −7.83308528800069718366669266159, −7.49854626403200308211384803397, −6.64015478034033294776010370075, −5.31166823001739669969266033554, −4.85834450825080168955307576161, −3.82761340677172253473232515863, −2.88210740590618426101881210417, −1.61933703544595924973302121161, 0,
1.61933703544595924973302121161, 2.88210740590618426101881210417, 3.82761340677172253473232515863, 4.85834450825080168955307576161, 5.31166823001739669969266033554, 6.64015478034033294776010370075, 7.49854626403200308211384803397, 7.83308528800069718366669266159, 8.903536245670668621130507435375
