L(s) = 1 | + (2.40 − 4.16i)2-s + (−1.5 − 2.59i)3-s + (−7.55 − 13.0i)4-s + (−2.5 + 4.33i)5-s − 14.4·6-s + (−18.4 − 0.916i)7-s − 34.1·8-s + (−4.5 + 7.79i)9-s + (12.0 + 20.8i)10-s + (−26.6 − 46.1i)11-s + (−22.6 + 39.2i)12-s + 60.9·13-s + (−48.2 + 74.7i)14-s + 15.0·15-s + (−21.6 + 37.4i)16-s + (−15.5 − 26.9i)17-s + ⋯ |
L(s) = 1 | + (0.849 − 1.47i)2-s + (−0.288 − 0.499i)3-s + (−0.943 − 1.63i)4-s + (−0.223 + 0.387i)5-s − 0.981·6-s + (−0.998 − 0.0494i)7-s − 1.50·8-s + (−0.166 + 0.288i)9-s + (0.379 + 0.658i)10-s + (−0.729 − 1.26i)11-s + (−0.544 + 0.943i)12-s + 1.30·13-s + (−0.921 + 1.42i)14-s + 0.258·15-s + (−0.337 + 0.585i)16-s + (−0.221 − 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.377669 + 1.44965i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.377669 + 1.44965i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 2.59i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (18.4 + 0.916i)T \) |
good | 2 | \( 1 + (-2.40 + 4.16i)T + (-4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (26.6 + 46.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 60.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + (15.5 + 26.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-8.95 + 15.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-17.4 + 30.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (58.8 + 101. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (87.6 - 151. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 411.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-145. + 251. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (291. + 504. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-328. - 569. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-208. + 361. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-283. - 491. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 887.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-546. - 946. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (67.7 - 117. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 464.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (15.9 - 27.6i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 254.T + 9.12e5T^{2} \) |
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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
−12.77765631489496752758317992951, −11.53680685651777820126431266142, −10.97206925789305312520573224802, −9.979515328718235859972409937125, −8.453092919080817799711737403654, −6.59990432187918655558864940167, −5.47938895223218035768753701164, −3.70739930034245205155414830905, −2.72021712981761545904342377002, −0.67125472143038036290580231823,
3.62019457042287640956862342666, 4.75716380527812804619769674177, 5.89233683105249080949113231565, 6.88639463329584295293308659203, 8.133086704706312509812816400832, 9.334366802545653214089285992936, 10.66466468041850964239655199538, 12.39388816685275620276085056648, 12.96676785809830311999938226268, 14.00408784802067505109338326227