| L(s) = 1 | + 77.7i·5-s − 249.·7-s − 2.46e3i·11-s + 504·13-s + 2.77e3i·17-s + 6.98e3·19-s + 1.23e4i·23-s + 9.57e3·25-s + 2.13e4i·29-s − 5.23e3·31-s − 1.93e4i·35-s − 3.78e4·37-s − 2.62e4i·41-s − 1.08e5·43-s − 1.72e4i·47-s + ⋯ |
| L(s) = 1 | + 0.622i·5-s − 0.727·7-s − 1.85i·11-s + 0.229·13-s + 0.564i·17-s + 1.01·19-s + 1.01i·23-s + 0.612·25-s + 0.873i·29-s − 0.175·31-s − 0.452i·35-s − 0.746·37-s − 0.381i·41-s − 1.36·43-s − 0.166i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6340380050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6340380050\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 77.7iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 249.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 2.46e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 504T + 4.82e6T^{2} \) |
| 17 | \( 1 - 2.77e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 6.98e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.23e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.13e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 5.23e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.78e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 2.62e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.08e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.72e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 4.48e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.53e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.93e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.73e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.60e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.89e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 9.58e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 4.91e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 9.22e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.52e6T + 8.32e11T^{2} \) |
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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
−11.06736025034172740626967924583, −10.35092176147723355597535294164, −9.222987586580747198483885785701, −8.361180983940556564490704514966, −7.16518951625974573401350507926, −6.22373268624511783902319928684, −5.36954809445636766944610491868, −3.52334597534757930654648222734, −3.07939804695267762585435534663, −1.24110426086958319680851422625,
0.16125032756029241918581659310, 1.57824732177666639133303481221, 2.92786326738147928127939690100, 4.35666528653930050989086235047, 5.17971505726378626685593115992, 6.57361924283853613123999451882, 7.37153597072835220154746664156, 8.565884093735293669141888980488, 9.626213379193776428999029083344, 10.06242165382432879733591117752
