L(s) = 1 | − 2-s + 3-s + 4-s − 3.09·5-s − 6-s − 7-s − 8-s + 9-s + 3.09·10-s − 6.22·11-s + 12-s + 5.60·13-s + 14-s − 3.09·15-s + 16-s + 6.87·17-s − 18-s − 4.45·19-s − 3.09·20-s − 21-s + 6.22·22-s − 4.13·23-s − 24-s + 4.59·25-s − 5.60·26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.38·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.979·10-s − 1.87·11-s + 0.288·12-s + 1.55·13-s + 0.267·14-s − 0.799·15-s + 0.250·16-s + 1.66·17-s − 0.235·18-s − 1.02·19-s − 0.692·20-s − 0.218·21-s + 1.32·22-s − 0.863·23-s − 0.204·24-s + 0.918·25-s − 1.10·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7927176242\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7927176242\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
good | 5 | \( 1 + 3.09T + 5T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 + 8.35T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 - 2.66T + 43T^{2} \) |
| 47 | \( 1 + 0.621T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 + 6.61T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 - 1.18T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 3.59T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 7.18T + 89T^{2} \) |
| 97 | \( 1 + 1.06T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.948599609662808237246489131785, −7.53240524276552832505802675648, −6.66336170563809024176162743200, −5.86562462845122725999976253169, −5.03754704379731160107990801432, −3.97240074082384514587144696818, −3.41528921485024610156102835069, −2.81215477000411907554540679360, −1.67669698967836121750306855150, −0.47066382186925288641462210533,
0.47066382186925288641462210533, 1.67669698967836121750306855150, 2.81215477000411907554540679360, 3.41528921485024610156102835069, 3.97240074082384514587144696818, 5.03754704379731160107990801432, 5.86562462845122725999976253169, 6.66336170563809024176162743200, 7.53240524276552832505802675648, 7.948599609662808237246489131785