L(s) = 1 | + (−14.7 + 5.18i)3-s − 32·4-s + 61.6i·5-s + 145.·7-s + (189. − 152. i)9-s + (470. − 165. i)12-s + (−319. − 906. i)15-s + 1.02e3·16-s − 683. i·17-s + 2.16e3·19-s − 1.97e3i·20-s + (−2.13e3 + 754. i)21-s − 679.·25-s + (−1.99e3 + 3.22e3i)27-s − 4.65e3·28-s + 5.70e3i·29-s + ⋯ |
L(s) = 1 | + (−0.943 + 0.332i)3-s − 4-s + 1.10i·5-s + 1.12·7-s + (0.778 − 0.627i)9-s + (0.943 − 0.332i)12-s + (−0.367 − 1.04i)15-s + 16-s − 0.573i·17-s + 1.37·19-s − 1.10i·20-s + (−1.05 + 0.373i)21-s − 0.217·25-s + (−0.525 + 0.850i)27-s − 1.12·28-s + 1.25i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.105145556\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105145556\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (14.7 - 5.18i)T \) |
| 59 | \( 1 + 2.67e4iT \) |
good | 2 | \( 1 + 32T^{2} \) |
| 5 | \( 1 - 61.6iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 145.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 - 3.71e5T^{2} \) |
| 17 | \( 1 + 683. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.16e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.70e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.83e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.35e3iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.94e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.55e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.58e9T^{2} \) |
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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
−11.81689678836557939717594120177, −11.13278095846932941562438845674, −10.21582722668111788408469155808, −9.310506208010546241398554059169, −7.913742721162134966829278932034, −6.84422700201162113832467607545, −5.43893393923653429228546132682, −4.69371959999365124516558447782, −3.31704531904092638713121883553, −1.12485488483231663446804191111,
0.53778461490728733386474666220, 1.50399233522670350682208692355, 4.19166889560640835876225326701, 5.02722651034342416519027942209, 5.73533240695052914924884328931, 7.55104593764214280159974916458, 8.379896163922879331173722680899, 9.424295318696453234877581049103, 10.55994238205281833638874061025, 11.73997553637135285736168915765