L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 1.01·7-s − 8-s + 9-s + 10-s − 11-s − 12-s + 5.50·13-s + 1.01·14-s + 15-s + 16-s + 2.68·17-s − 18-s + 19-s − 20-s + 1.01·21-s + 22-s − 8.59·23-s + 24-s + 25-s − 5.50·26-s − 27-s − 1.01·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 0.383·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 1.52·13-s + 0.271·14-s + 0.258·15-s + 0.250·16-s + 0.651·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.221·21-s + 0.213·22-s − 1.79·23-s + 0.204·24-s + 0.200·25-s − 1.07·26-s − 0.192·27-s − 0.191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9135369178\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9135369178\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 1.01T + 7T^{2} \) |
| 13 | \( 1 - 5.50T + 13T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 23 | \( 1 + 8.59T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 - 6.95T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 - 1.71T + 61T^{2} \) |
| 67 | \( 1 - 2.32T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 0.0870T + 83T^{2} \) |
| 89 | \( 1 - 8.55T + 89T^{2} \) |
| 97 | \( 1 + 3.17T + 97T^{2} \) |
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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
−7.971885478886950351237195585871, −7.58368907398203590917866539237, −6.43269852701500120186208352679, −6.19249068703246041168008334807, −5.40586451897269801994810871291, −4.29425516787544787621384833886, −3.64691907162484875680319810513, −2.72061555580207576840975418627, −1.51946422362523121628762332776, −0.60152415948064867737394383275,
0.60152415948064867737394383275, 1.51946422362523121628762332776, 2.72061555580207576840975418627, 3.64691907162484875680319810513, 4.29425516787544787621384833886, 5.40586451897269801994810871291, 6.19249068703246041168008334807, 6.43269852701500120186208352679, 7.58368907398203590917866539237, 7.971885478886950351237195585871