L(s) = 1 | + 2-s + 4-s − 3·5-s + 3·7-s + 8-s − 3·10-s + 3·14-s + 16-s + 3·17-s + 6·19-s − 3·20-s − 6·23-s + 4·25-s + 3·28-s + 32-s + 3·34-s − 9·35-s + 3·37-s + 6·38-s − 3·40-s + 43-s − 6·46-s − 3·47-s + 2·49-s + 4·50-s + 6·53-s + 3·56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 1.13·7-s + 0.353·8-s − 0.948·10-s + 0.801·14-s + 1/4·16-s + 0.727·17-s + 1.37·19-s − 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.566·28-s + 0.176·32-s + 0.514·34-s − 1.52·35-s + 0.493·37-s + 0.973·38-s − 0.474·40-s + 0.152·43-s − 0.884·46-s − 0.437·47-s + 2/7·49-s + 0.565·50-s + 0.824·53-s + 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.703869333\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.703869333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.331440703527111351101171793431, −7.82483480853824997311953087544, −7.46416402975249861419015857646, −6.41880784059227063665504592677, −5.39125907693327786955304301989, −4.84993076407059258419808561866, −3.93579076010265315483518705314, −3.42461604885341242516757162261, −2.18700281733218872733800900597, −0.935116836164402058512878230992,
0.935116836164402058512878230992, 2.18700281733218872733800900597, 3.42461604885341242516757162261, 3.93579076010265315483518705314, 4.84993076407059258419808561866, 5.39125907693327786955304301989, 6.41880784059227063665504592677, 7.46416402975249861419015857646, 7.82483480853824997311953087544, 8.331440703527111351101171793431