L(s) = 1 | + 2-s − 1.62·3-s + 4-s + 5-s − 1.62·6-s − 0.333·7-s + 8-s − 0.360·9-s + 10-s − 2.68·11-s − 1.62·12-s + 1.45·13-s − 0.333·14-s − 1.62·15-s + 16-s − 4.11·17-s − 0.360·18-s + 8.34·19-s + 20-s + 0.541·21-s − 2.68·22-s − 4.74·23-s − 1.62·24-s + 25-s + 1.45·26-s + 5.45·27-s − 0.333·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.938·3-s + 0.5·4-s + 0.447·5-s − 0.663·6-s − 0.125·7-s + 0.353·8-s − 0.120·9-s + 0.316·10-s − 0.810·11-s − 0.469·12-s + 0.403·13-s − 0.0890·14-s − 0.419·15-s + 0.250·16-s − 0.996·17-s − 0.0849·18-s + 1.91·19-s + 0.223·20-s + 0.118·21-s − 0.573·22-s − 0.989·23-s − 0.331·24-s + 0.200·25-s + 0.285·26-s + 1.05·27-s − 0.0629·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 1.62T + 3T^{2} \) |
| 7 | \( 1 + 0.333T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 - 8.34T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 + 3.92T + 31T^{2} \) |
| 37 | \( 1 - 3.53T + 37T^{2} \) |
| 41 | \( 1 - 2.16T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 59 | \( 1 + 3.43T + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 - 0.556T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.81T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.59648389815351550155590827734, −6.71801407033005821713430717270, −6.16364251016455415779615506244, −5.45100531192964046899030532898, −5.13990456866016228069942845728, −4.19948469011988128294419845811, −3.23033800402537587947912405028, −2.46901537625911401857021284938, −1.35764252188763488969668886337, 0,
1.35764252188763488969668886337, 2.46901537625911401857021284938, 3.23033800402537587947912405028, 4.19948469011988128294419845811, 5.13990456866016228069942845728, 5.45100531192964046899030532898, 6.16364251016455415779615506244, 6.71801407033005821713430717270, 7.59648389815351550155590827734