| L(s) = 1 | + 2-s − 1.74·3-s + 4-s + 2.79·5-s − 1.74·6-s + 0.229·7-s + 8-s + 0.0613·9-s + 2.79·10-s − 4.72·11-s − 1.74·12-s + 0.543·13-s + 0.229·14-s − 4.89·15-s + 16-s − 1.47·17-s + 0.0613·18-s − 0.554·19-s + 2.79·20-s − 0.401·21-s − 4.72·22-s − 23-s − 1.74·24-s + 2.83·25-s + 0.543·26-s + 5.14·27-s + 0.229·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.01·3-s + 0.5·4-s + 1.25·5-s − 0.714·6-s + 0.0867·7-s + 0.353·8-s + 0.0204·9-s + 0.885·10-s − 1.42·11-s − 0.505·12-s + 0.150·13-s + 0.0613·14-s − 1.26·15-s + 0.250·16-s − 0.356·17-s + 0.0144·18-s − 0.127·19-s + 0.626·20-s − 0.0876·21-s − 1.00·22-s − 0.208·23-s − 0.357·24-s + 0.567·25-s + 0.106·26-s + 0.989·27-s + 0.0433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 - 0.229T + 7T^{2} \) |
| 11 | \( 1 + 4.72T + 11T^{2} \) |
| 13 | \( 1 - 0.543T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 0.554T + 19T^{2} \) |
| 29 | \( 1 + 3.35T + 29T^{2} \) |
| 31 | \( 1 - 7.40T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 43 | \( 1 - 9.53T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 2.44T + 61T^{2} \) |
| 67 | \( 1 + 7.79T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 3.82T + 79T^{2} \) |
| 83 | \( 1 + 0.949T + 83T^{2} \) |
| 89 | \( 1 - 7.74T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.57268344550967789398852270531, −6.61750381180497626774100188318, −6.12320212603952524819251783940, −5.54583630915102713750429577973, −5.07487542888636429923952007295, −4.37958510787215722953884015403, −3.07354085099696283205545034039, −2.41710023038497112032977294492, −1.47061521295657132776924561592, 0,
1.47061521295657132776924561592, 2.41710023038497112032977294492, 3.07354085099696283205545034039, 4.37958510787215722953884015403, 5.07487542888636429923952007295, 5.54583630915102713750429577973, 6.12320212603952524819251783940, 6.61750381180497626774100188318, 7.57268344550967789398852270531
