| L(s) = 1 | + (−5.78 + 7.95i)3-s + (10.1 + 4.71i)5-s + 24.7i·7-s + (−21.5 − 66.3i)9-s + (1.97 − 6.07i)11-s + (−62.0 + 20.1i)13-s + (−96.1 + 53.3i)15-s + (−35.7 − 49.2i)17-s + (43.8 − 31.8i)19-s + (−196. − 143. i)21-s + (7.45 + 2.42i)23-s + (80.4 + 95.6i)25-s + (400. + 130. i)27-s + (105. + 76.4i)29-s + (−231. + 168. i)31-s + ⋯ |
| L(s) = 1 | + (−1.11 + 1.53i)3-s + (0.906 + 0.422i)5-s + 1.33i·7-s + (−0.798 − 2.45i)9-s + (0.0540 − 0.166i)11-s + (−1.32 + 0.429i)13-s + (−1.65 + 0.918i)15-s + (−0.510 − 0.702i)17-s + (0.529 − 0.384i)19-s + (−2.04 − 1.48i)21-s + (0.0675 + 0.0219i)23-s + (0.643 + 0.765i)25-s + (2.85 + 0.926i)27-s + (0.673 + 0.489i)29-s + (−1.34 + 0.975i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00827i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00358217 - 0.865829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00358217 - 0.865829i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-10.1 - 4.71i)T \) |
| good | 3 | \( 1 + (5.78 - 7.95i)T + (-8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 - 24.7iT - 343T^{2} \) |
| 11 | \( 1 + (-1.97 + 6.07i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (62.0 - 20.1i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (35.7 + 49.2i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-43.8 + 31.8i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-7.45 - 2.42i)T + (9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-105. - 76.4i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (231. - 168. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-237. + 77.1i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-18.4 - 56.7i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 32.9iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (339. - 467. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (0.813 - 1.11i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (19.0 + 58.6i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (185. - 571. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (357. + 491. i)T + (-9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-544. - 395. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-331. - 107. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (7.93 + 5.76i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-719. - 990. i)T + (-1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (248. - 764. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-528. + 726. i)T + (-2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.31609741393642306433375694550, −12.50911643709753817043803495272, −11.59338495161403871408473909512, −10.72470267189558768654103125295, −9.490468244874614592428630239010, −9.222367626364464057049959457307, −6.68268346596412712633285692325, −5.54467792075375987243540460625, −4.84019788982369933543693811866, −2.82981163853603424146144401814,
0.55016379134604358834996402531, 1.95198934464948595469441645140, 4.85813022442964141767000245041, 6.04158262123678028616792768044, 7.07367601142801113915338684190, 7.942099909182712165271121378382, 9.886237375559238178270797086702, 10.83054748524242106252606183979, 12.02420510522258372336070521445, 12.97239635541963406924542460467
