| L(s) = 1 | + (−24.1 − 12.0i)3-s − 55.9i·5-s − 436.·7-s + (437. + 582. i)9-s − 1.10e3i·11-s + 908.·13-s + (−674. + 1.35e3i)15-s + 9.74e3i·17-s + 7.70e3·19-s + (1.05e4 + 5.27e3i)21-s + 1.93e4i·23-s − 3.12e3·25-s + (−3.54e3 − 1.93e4i)27-s + 8.10e3i·29-s − 768.·31-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.446i)3-s − 0.447i·5-s − 1.27·7-s + (0.600 + 0.799i)9-s − 0.826i·11-s + 0.413·13-s + (−0.199 + 0.400i)15-s + 1.98i·17-s + 1.12·19-s + (1.13 + 0.569i)21-s + 1.58i·23-s − 0.199·25-s + (−0.180 − 0.983i)27-s + 0.332i·29-s − 0.0258·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.629151 + 0.389004i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.629151 + 0.389004i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (24.1 + 12.0i)T \) |
| 5 | \( 1 + 55.9iT \) |
| good | 7 | \( 1 + 436.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.10e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 908.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 9.74e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 7.70e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.93e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 8.10e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 768.T + 8.87e8T^{2} \) |
| 37 | \( 1 - 7.22e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.40e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.21e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 3.70e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.73e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.38e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.78e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 9.62e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.38e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.75e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.21e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 3.10e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.11e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.06e5T + 8.32e11T^{2} \) |
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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
−13.55899750614100576458030262579, −12.95951178133229385685372555076, −11.88148575831417852634359240255, −10.71388644206173168271920932968, −9.490345008932466236817359402515, −7.955982986463744863651310087788, −6.42638559792700349794279155104, −5.57809740344011642525275172668, −3.60340131299403986118512926727, −1.21648222412615385828720146839,
0.39412675499165162689238244083, 3.06469269988125463829101158811, 4.73373092861345247706637783987, 6.25805909352562939358995078723, 7.18885402769717599626926933631, 9.432143346797925464995110358285, 10.06217519137825132349213938036, 11.39719747359220453729548396088, 12.35470013088333332510897643619, 13.53752812559169356586350530211
