L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·8-s + 9-s − 2·10-s − 12-s + 6·13-s − 2·15-s − 16-s − 6·17-s + 18-s − 19-s + 2·20-s + 4·23-s − 3·24-s − 25-s + 6·26-s + 27-s + 2·29-s − 2·30-s + 8·31-s + 5·32-s − 6·34-s − 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 1.66·13-s − 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s + 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s − 1.02·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.085302272\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.085302272\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.35090650815041953121172131081, −13.89424674132429450373973021658, −13.30135307533368428997273937212, −12.07987440084703696834915246162, −10.84155613367776600825623791063, −9.032116632714081125535639739797, −8.263513432862467471567669978548, −6.46787589567348650226556085189, −4.56991321080597356703354311172, −3.44365518362789223021999993571,
3.44365518362789223021999993571, 4.56991321080597356703354311172, 6.46787589567348650226556085189, 8.263513432862467471567669978548, 9.032116632714081125535639739797, 10.84155613367776600825623791063, 12.07987440084703696834915246162, 13.30135307533368428997273937212, 13.89424674132429450373973021658, 15.35090650815041953121172131081