| L(s) = 1 | − 1.36·2-s − 6.14·4-s − 3.03·5-s − 7.94·7-s + 19.2·8-s + 4.12·10-s − 27.6·11-s + 58.1·13-s + 10.8·14-s + 22.9·16-s + 17·17-s + 89.1·19-s + 18.6·20-s + 37.5·22-s + 115.·23-s − 115.·25-s − 79.1·26-s + 48.8·28-s + 128.·29-s + 273.·31-s − 185.·32-s − 23.1·34-s + 24.0·35-s − 132.·37-s − 121.·38-s − 58.3·40-s + 470.·41-s + ⋯ |
| L(s) = 1 | − 0.481·2-s − 0.768·4-s − 0.271·5-s − 0.428·7-s + 0.851·8-s + 0.130·10-s − 0.756·11-s + 1.23·13-s + 0.206·14-s + 0.358·16-s + 0.242·17-s + 1.07·19-s + 0.208·20-s + 0.364·22-s + 1.04·23-s − 0.926·25-s − 0.596·26-s + 0.329·28-s + 0.823·29-s + 1.58·31-s − 1.02·32-s − 0.116·34-s + 0.116·35-s − 0.588·37-s − 0.518·38-s − 0.230·40-s + 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9558484037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9558484037\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 1.36T + 8T^{2} \) |
| 5 | \( 1 + 3.03T + 125T^{2} \) |
| 7 | \( 1 + 7.94T + 343T^{2} \) |
| 11 | \( 1 + 27.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.1T + 2.19e3T^{2} \) |
| 19 | \( 1 - 89.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 527.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 53.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 52.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 788.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 295.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 720.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 116.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 813.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 794.T + 9.12e5T^{2} \) |
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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
−12.66757292263409708323066742514, −11.35235431967553903999591595823, −10.31819639495361563011258468862, −9.415087799851179664143929882773, −8.396212568951396220914393036712, −7.52411789311406541898030271039, −5.96520668290534900983468579294, −4.64229333094379753346661223765, −3.25105324535129063392862491881, −0.888174854581830993379655277345,
0.888174854581830993379655277345, 3.25105324535129063392862491881, 4.64229333094379753346661223765, 5.96520668290534900983468579294, 7.52411789311406541898030271039, 8.396212568951396220914393036712, 9.415087799851179664143929882773, 10.31819639495361563011258468862, 11.35235431967553903999591595823, 12.66757292263409708323066742514
