L(s) = 1 | + 5·5-s + 18·7-s − 16·11-s − 6·13-s + 6·17-s + 124·19-s + 42·23-s + 25·25-s − 142·29-s + 188·31-s + 90·35-s + 202·37-s − 54·41-s − 66·43-s + 38·47-s − 19·49-s − 738·53-s − 80·55-s + 564·59-s − 262·61-s − 30·65-s + 554·67-s + 140·71-s + 882·73-s − 288·77-s + 1.16e3·79-s + 642·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.971·7-s − 0.438·11-s − 0.128·13-s + 0.0856·17-s + 1.49·19-s + 0.380·23-s + 1/5·25-s − 0.909·29-s + 1.08·31-s + 0.434·35-s + 0.897·37-s − 0.205·41-s − 0.234·43-s + 0.117·47-s − 0.0553·49-s − 1.91·53-s − 0.196·55-s + 1.24·59-s − 0.549·61-s − 0.0572·65-s + 1.01·67-s + 0.234·71-s + 1.41·73-s − 0.426·77-s + 1.65·79-s + 0.849·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.567030314\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.567030314\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
good | 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 142 T + p^{3} T^{2} \) |
| 31 | \( 1 - 188 T + p^{3} T^{2} \) |
| 37 | \( 1 - 202 T + p^{3} T^{2} \) |
| 41 | \( 1 + 54 T + p^{3} T^{2} \) |
| 43 | \( 1 + 66 T + p^{3} T^{2} \) |
| 47 | \( 1 - 38 T + p^{3} T^{2} \) |
| 53 | \( 1 + 738 T + p^{3} T^{2} \) |
| 59 | \( 1 - 564 T + p^{3} T^{2} \) |
| 61 | \( 1 + 262 T + p^{3} T^{2} \) |
| 67 | \( 1 - 554 T + p^{3} T^{2} \) |
| 71 | \( 1 - 140 T + p^{3} T^{2} \) |
| 73 | \( 1 - 882 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 642 T + p^{3} T^{2} \) |
| 89 | \( 1 - 854 T + p^{3} T^{2} \) |
| 97 | \( 1 + 478 T + p^{3} T^{2} \) |
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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
−9.920849793136753121959957260672, −9.287861565403620424961876351027, −8.137261167631183614528329892314, −7.58929732362836523526354771533, −6.45364988355349428546350124599, −5.35454170548207915294112014857, −4.77263737690471452286913231279, −3.35635530913369842802878880504, −2.15682645795474749058403339358, −0.965400042184855393392491537749,
0.965400042184855393392491537749, 2.15682645795474749058403339358, 3.35635530913369842802878880504, 4.77263737690471452286913231279, 5.35454170548207915294112014857, 6.45364988355349428546350124599, 7.58929732362836523526354771533, 8.137261167631183614528329892314, 9.287861565403620424961876351027, 9.920849793136753121959957260672