| L(s) = 1 | + 71·7-s + 581.·11-s + 137·13-s − 581.·17-s − 1.08e3·19-s − 4.07e3·23-s − 4.07e3·29-s − 2.26e3·31-s − 6.01e3·37-s + 1.74e3·41-s + 4.28e3·43-s − 1.16e3·47-s − 1.17e4·49-s + 2.50e4·53-s + 2.90e4·59-s + 1.27e4·61-s + 6.89e3·67-s − 4.01e4·71-s − 2.40e4·73-s + 4.13e4·77-s + 1.22e4·79-s − 2.67e4·83-s − 5.58e4·89-s + 9.72e3·91-s − 1.96e4·97-s + 1.36e5·101-s − 1.52e5·103-s + ⋯ |
| L(s) = 1 | + 0.547·7-s + 1.44·11-s + 0.224·13-s − 0.488·17-s − 0.690·19-s − 1.60·23-s − 0.899·29-s − 0.424·31-s − 0.721·37-s + 0.162·41-s + 0.353·43-s − 0.0768·47-s − 0.700·49-s + 1.22·53-s + 1.08·59-s + 0.437·61-s + 0.187·67-s − 0.944·71-s − 0.527·73-s + 0.793·77-s + 0.220·79-s − 0.426·83-s − 0.747·89-s + 0.123·91-s − 0.212·97-s + 1.33·101-s − 1.41·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 71T + 1.68e4T^{2} \) |
| 11 | \( 1 - 581.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 137T + 3.71e5T^{2} \) |
| 17 | \( 1 + 581.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.08e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.74e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.16e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.50e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.89e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.96e4T + 8.58e9T^{2} \) |
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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
−8.888954799435733716167079052009, −8.239226161394713768101235134473, −7.18951146433282316023162472139, −6.37873282359787983816503002457, −5.52914387086260051031458039337, −4.28240873464275319575171832451, −3.75036968853651940800335791096, −2.19920863519655429581735920739, −1.37866151040090094471565193068, 0,
1.37866151040090094471565193068, 2.19920863519655429581735920739, 3.75036968853651940800335791096, 4.28240873464275319575171832451, 5.52914387086260051031458039337, 6.37873282359787983816503002457, 7.18951146433282316023162472139, 8.239226161394713768101235134473, 8.888954799435733716167079052009
