| L(s) = 1 | + 2-s + 4-s − 3.85·5-s + 0.618·7-s + 8-s − 3·9-s − 3.85·10-s + 2.47·13-s + 0.618·14-s + 16-s + 4.09·17-s − 3·18-s + 19-s − 3.85·20-s + 1.14·23-s + 9.85·25-s + 2.47·26-s + 0.618·28-s + 1.23·29-s − 5.70·31-s + 32-s + 4.09·34-s − 2.38·35-s − 3·36-s − 9.23·37-s + 38-s − 3.85·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.72·5-s + 0.233·7-s + 0.353·8-s − 9-s − 1.21·10-s + 0.685·13-s + 0.165·14-s + 0.250·16-s + 0.992·17-s − 0.707·18-s + 0.229·19-s − 0.861·20-s + 0.238·23-s + 1.97·25-s + 0.484·26-s + 0.116·28-s + 0.229·29-s − 1.02·31-s + 0.176·32-s + 0.701·34-s − 0.402·35-s − 0.5·36-s − 1.51·37-s + 0.162·38-s − 0.609·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 - 4.09T + 17T^{2} \) |
| 23 | \( 1 - 1.14T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 8.47T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 3.85T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 7.52T + 97T^{2} \) |
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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
−7.922423084051377199441567525225, −7.31228232469669571200829764971, −6.50073216225021836768639610536, −5.58614965853534317957821843811, −4.97458200674811385571382107123, −4.05070849024672284493975996004, −3.44027947270353316130584127356, −2.88351527594001318963969959242, −1.36576828426567472287871373116, 0,
1.36576828426567472287871373116, 2.88351527594001318963969959242, 3.44027947270353316130584127356, 4.05070849024672284493975996004, 4.97458200674811385571382107123, 5.58614965853534317957821843811, 6.50073216225021836768639610536, 7.31228232469669571200829764971, 7.922423084051377199441567525225
