L(s) = 1 | − 2-s − 3-s + 4-s + 1.77·5-s + 6-s − 3.99·7-s − 8-s + 9-s − 1.77·10-s − 0.580·11-s − 12-s + 13-s + 3.99·14-s − 1.77·15-s + 16-s + 6.41·17-s − 18-s − 4.99·19-s + 1.77·20-s + 3.99·21-s + 0.580·22-s − 6.59·23-s + 24-s − 1.84·25-s − 26-s − 27-s − 3.99·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.794·5-s + 0.408·6-s − 1.50·7-s − 0.353·8-s + 0.333·9-s − 0.561·10-s − 0.174·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s − 0.458·15-s + 0.250·16-s + 1.55·17-s − 0.235·18-s − 1.14·19-s + 0.397·20-s + 0.871·21-s + 0.123·22-s − 1.37·23-s + 0.204·24-s − 0.368·25-s − 0.196·26-s − 0.192·27-s − 0.754·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 + 0.580T + 11T^{2} \) |
| 17 | \( 1 - 6.41T + 17T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 1.31T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 - 6.39T + 37T^{2} \) |
| 41 | \( 1 - 6.04T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 6.62T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 2.66T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 - 7.90T + 83T^{2} \) |
| 89 | \( 1 + 3.95T + 89T^{2} \) |
| 97 | \( 1 - 8.16T + 97T^{2} \) |
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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
−7.71830054067671860479979804770, −6.48792261159569398382976250081, −6.05552756325167038200323892037, −5.99713332775557076293989844627, −4.74044710666842190976505124647, −3.78255092591082478791654089450, −2.95321217729174614152826210791, −2.13309196936108115146503501238, −1.05626427956969767782349388641, 0,
1.05626427956969767782349388641, 2.13309196936108115146503501238, 2.95321217729174614152826210791, 3.78255092591082478791654089450, 4.74044710666842190976505124647, 5.99713332775557076293989844627, 6.05552756325167038200323892037, 6.48792261159569398382976250081, 7.71830054067671860479979804770