L(s) = 1 | + 3.16·3-s − 2.87·5-s − 4.05·7-s + 7.04·9-s − 0.0722·11-s − 0.263·13-s − 9.12·15-s + 3.42·17-s + 19-s − 12.8·21-s + 2.37·23-s + 3.28·25-s + 12.8·27-s + 1.44·29-s − 9.00·31-s − 0.228·33-s + 11.6·35-s − 7.21·37-s − 0.835·39-s + 6.92·41-s − 8.85·43-s − 20.2·45-s − 1.76·47-s + 9.44·49-s + 10.8·51-s + 3.29·53-s + 0.207·55-s + ⋯ |
L(s) = 1 | + 1.83·3-s − 1.28·5-s − 1.53·7-s + 2.34·9-s − 0.0217·11-s − 0.0730·13-s − 2.35·15-s + 0.831·17-s + 0.229·19-s − 2.80·21-s + 0.496·23-s + 0.657·25-s + 2.46·27-s + 0.269·29-s − 1.61·31-s − 0.0398·33-s + 1.97·35-s − 1.18·37-s − 0.133·39-s + 1.08·41-s − 1.35·43-s − 3.02·45-s − 0.257·47-s + 1.34·49-s + 1.52·51-s + 0.453·53-s + 0.0280·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 + 0.0722T + 11T^{2} \) |
| 13 | \( 1 + 0.263T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 8.85T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 - 0.351T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 - 1.30T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 83 | \( 1 + 0.269T + 83T^{2} \) |
| 89 | \( 1 + 7.38T + 89T^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−7.72529023087375212279256453685, −7.24952065076157054340406017913, −6.72791135260303154518255267644, −5.55930251027559376453731500594, −4.39734587369513393480705419307, −3.69484874633191472476422622372, −3.27436743109074801673922871782, −2.75523468375582017426261890934, −1.48292504214974470030995127225, 0,
1.48292504214974470030995127225, 2.75523468375582017426261890934, 3.27436743109074801673922871782, 3.69484874633191472476422622372, 4.39734587369513393480705419307, 5.55930251027559376453731500594, 6.72791135260303154518255267644, 7.24952065076157054340406017913, 7.72529023087375212279256453685