| L(s) = 1 | + 2-s + 3-s + 4-s − 3.11·5-s + 6-s − 0.395·7-s + 8-s + 9-s − 3.11·10-s − 4.45·11-s + 12-s + 13-s − 0.395·14-s − 3.11·15-s + 16-s − 3.51·17-s + 18-s − 1.39·19-s − 3.11·20-s − 0.395·21-s − 4.45·22-s + 8.68·23-s + 24-s + 4.71·25-s + 26-s + 27-s − 0.395·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.39·5-s + 0.408·6-s − 0.149·7-s + 0.353·8-s + 0.333·9-s − 0.985·10-s − 1.34·11-s + 0.288·12-s + 0.277·13-s − 0.105·14-s − 0.804·15-s + 0.250·16-s − 0.852·17-s + 0.235·18-s − 0.320·19-s − 0.697·20-s − 0.0863·21-s − 0.949·22-s + 1.81·23-s + 0.204·24-s + 0.943·25-s + 0.196·26-s + 0.192·27-s − 0.0747·28-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
| \(L(1)\) |
\(\approx\) |
\(2.422039485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.422039485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.11T + 5T^{2} \) |
| 7 | \( 1 + 0.395T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 + 1.39T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 + 7.36T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 - 2.61T + 43T^{2} \) |
| 47 | \( 1 - 7.31T + 47T^{2} \) |
| 53 | \( 1 - 8.42T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 - 6.69T + 67T^{2} \) |
| 71 | \( 1 - 9.60T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 9.52T + 97T^{2} \) |
| 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
−7.81459018355006406601537431953, −6.99240767748862448753376624177, −6.79625464284104501367494554408, −5.52110548492715173950336362645, −4.85800730393802710612879934061, −4.28296411439997779731674717253, −3.43759064672298331495371532298, −2.95975294519506582062820976320, −2.09862324609326522058792227748, −0.64622012925829370038428128537,
0.64622012925829370038428128537, 2.09862324609326522058792227748, 2.95975294519506582062820976320, 3.43759064672298331495371532298, 4.28296411439997779731674717253, 4.85800730393802710612879934061, 5.52110548492715173950336362645, 6.79625464284104501367494554408, 6.99240767748862448753376624177, 7.81459018355006406601537431953
