L(s) = 1 | + 1.73·2-s − 3-s + 0.999·4-s − 1.73·6-s − 2·7-s − 1.73·8-s + 9-s − 11-s − 0.999·12-s − 5.46·13-s − 3.46·14-s − 5·16-s + 1.73·18-s + 5.46·19-s + 2·21-s − 1.73·22-s − 6.92·23-s + 1.73·24-s − 9.46·26-s − 27-s − 1.99·28-s − 3.46·29-s − 10.9·31-s − 5.19·32-s + 33-s + 0.999·36-s + 4.92·37-s + ⋯ |
L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.707·6-s − 0.755·7-s − 0.612·8-s + 0.333·9-s − 0.301·11-s − 0.288·12-s − 1.51·13-s − 0.925·14-s − 1.25·16-s + 0.408·18-s + 1.25·19-s + 0.436·21-s − 0.369·22-s − 1.44·23-s + 0.353·24-s − 1.85·26-s − 0.192·27-s − 0.377·28-s − 0.643·29-s − 1.96·31-s − 0.918·32-s + 0.174·33-s + 0.166·36-s + 0.810·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 0.928T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−9.686714701389686651026132086190, −9.370474085062089692376184232137, −7.72974972916437071272652813370, −7.01502307853168229988256702430, −5.89127471605424311936608355157, −5.42177654495458161759189796888, −4.43066430794526292534749169160, −3.49179412737773871035965089826, −2.37648288046462023441843942037, 0,
2.37648288046462023441843942037, 3.49179412737773871035965089826, 4.43066430794526292534749169160, 5.42177654495458161759189796888, 5.89127471605424311936608355157, 7.01502307853168229988256702430, 7.72974972916437071272652813370, 9.370474085062089692376184232137, 9.686714701389686651026132086190