| L(s) = 1 | + 2-s + 2.74·3-s + 4-s + 1.49·5-s + 2.74·6-s − 4.04·7-s + 8-s + 4.52·9-s + 1.49·10-s − 0.488·11-s + 2.74·12-s − 2.89·13-s − 4.04·14-s + 4.11·15-s + 16-s + 2.15·17-s + 4.52·18-s + 8.01·19-s + 1.49·20-s − 11.0·21-s − 0.488·22-s − 23-s + 2.74·24-s − 2.75·25-s − 2.89·26-s + 4.19·27-s − 4.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.58·3-s + 0.5·4-s + 0.669·5-s + 1.12·6-s − 1.52·7-s + 0.353·8-s + 1.50·9-s + 0.473·10-s − 0.147·11-s + 0.792·12-s − 0.801·13-s − 1.08·14-s + 1.06·15-s + 0.250·16-s + 0.522·17-s + 1.06·18-s + 1.83·19-s + 0.334·20-s − 2.41·21-s − 0.104·22-s − 0.208·23-s + 0.560·24-s − 0.551·25-s − 0.566·26-s + 0.807·27-s − 0.763·28-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
| \(L(1)\) |
\(\approx\) |
\(5.857787921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.857787921\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.74T + 3T^{2} \) |
| 5 | \( 1 - 1.49T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 + 0.488T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 8.01T + 19T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 + 1.81T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 1.37T + 67T^{2} \) |
| 71 | \( 1 + 5.28T + 71T^{2} \) |
| 73 | \( 1 + 6.54T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 7.19T + 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
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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
−7.905092341589624939796814588176, −7.44986336654982726082220388952, −6.64608296497179138141918014556, −5.99735127621247244790199024271, −5.17593077590063787046164944700, −4.20891659613502628556642078989, −3.40045052134610826354939105083, −2.78676242095512220117128044265, −2.46220186247439998471386329696, −1.10105564167779802894382300376,
1.10105564167779802894382300376, 2.46220186247439998471386329696, 2.78676242095512220117128044265, 3.40045052134610826354939105083, 4.20891659613502628556642078989, 5.17593077590063787046164944700, 5.99735127621247244790199024271, 6.64608296497179138141918014556, 7.44986336654982726082220388952, 7.905092341589624939796814588176
