| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4508268022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4508268022\) |
| \(L(4)\) |
|
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
−10.24786837537581660692528725260, −9.061981115614077791957920670093, −8.476306666931575217066040292171, −7.27516897054829183515764247315, −6.40871411403686515321440775915, −5.76681430590320348368669018510, −4.39940513129105727676214631928, −3.20463031373302397344095668506, −2.63260226540869282106466069746, −0.888174954811882945545493400178,
0.10711200979032827579159436378, 1.52521580907217033248323050099, 2.51441496333230246414665245017, 3.97061575165490357177724253064, 4.68282136369402438313505040379, 5.80701492561055769177136441829, 6.82394301826479941960425784117, 7.61335227193889454311433239429, 8.676471635629472014096061360851, 9.596975829003893004759787908534
