L(s) = 1 | − 2-s − 2.76·3-s + 4-s − 5-s + 2.76·6-s − 3.85·7-s − 8-s + 4.66·9-s + 10-s − 1.48·11-s − 2.76·12-s + 13-s + 3.85·14-s + 2.76·15-s + 16-s + 7.93·17-s − 4.66·18-s + 4.98·19-s − 20-s + 10.6·21-s + 1.48·22-s − 2.13·23-s + 2.76·24-s + 25-s − 26-s − 4.59·27-s − 3.85·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.59·3-s + 0.5·4-s − 0.447·5-s + 1.12·6-s − 1.45·7-s − 0.353·8-s + 1.55·9-s + 0.316·10-s − 0.448·11-s − 0.798·12-s + 0.277·13-s + 1.03·14-s + 0.714·15-s + 0.250·16-s + 1.92·17-s − 1.09·18-s + 1.14·19-s − 0.223·20-s + 2.33·21-s + 0.316·22-s − 0.444·23-s + 0.564·24-s + 0.200·25-s − 0.196·26-s − 0.884·27-s − 0.729·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3014447206\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3014447206\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 17 | \( 1 - 7.93T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 + 2.13T + 23T^{2} \) |
| 29 | \( 1 + 5.71T + 29T^{2} \) |
| 37 | \( 1 + 9.26T + 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 3.00T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 8.84T + 67T^{2} \) |
| 71 | \( 1 + 2.57T + 71T^{2} \) |
| 73 | \( 1 - 1.37T + 73T^{2} \) |
| 79 | \( 1 + 3.05T + 79T^{2} \) |
| 83 | \( 1 + 0.685T + 83T^{2} \) |
| 89 | \( 1 - 3.35T + 89T^{2} \) |
| 97 | \( 1 - 2.50T + 97T^{2} \) |
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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
−8.367775462177420076369651205307, −7.47782873448308783036077724840, −7.04629321727647098794828190964, −6.17806239995175353368745988307, −5.66430471856100738355138315004, −5.03070789030828094453679880019, −3.63187684877022461811187637355, −3.14798938253438943711405046822, −1.45335917854309949876132019772, −0.39925239208154378160620553412,
0.39925239208154378160620553412, 1.45335917854309949876132019772, 3.14798938253438943711405046822, 3.63187684877022461811187637355, 5.03070789030828094453679880019, 5.66430471856100738355138315004, 6.17806239995175353368745988307, 7.04629321727647098794828190964, 7.47782873448308783036077724840, 8.367775462177420076369651205307