| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 6.00i·5-s + 4.69i·7-s − 2.82·8-s + 8.49i·10-s − 6.16i·11-s + 8.71·13-s − 6.63i·14-s + 4.00·16-s − 14.0i·17-s − 14.6i·19-s − 12.0i·20-s + 8.72i·22-s + (−9.92 + 20.7i)23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 1.20i·5-s + 0.670i·7-s − 0.353·8-s + 0.849i·10-s − 0.560i·11-s + 0.670·13-s − 0.473i·14-s + 0.250·16-s − 0.828i·17-s − 0.770i·19-s − 0.600i·20-s + 0.396i·22-s + (−0.431 + 0.902i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.482655 - 0.765833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.482655 - 0.765833i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (9.92 - 20.7i)T \) |
| good | 5 | \( 1 + 6.00iT - 25T^{2} \) |
| 7 | \( 1 - 4.69iT - 49T^{2} \) |
| 11 | \( 1 + 6.16iT - 121T^{2} \) |
| 13 | \( 1 - 8.71T + 169T^{2} \) |
| 17 | \( 1 + 14.0iT - 289T^{2} \) |
| 19 | \( 1 + 14.6iT - 361T^{2} \) |
| 29 | \( 1 + 1.08T + 841T^{2} \) |
| 31 | \( 1 + 48.1T + 961T^{2} \) |
| 37 | \( 1 + 56.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 22.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 47.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 7.39T + 2.20e3T^{2} \) |
| 53 | \( 1 + 86.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 87.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 0.0196iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 104. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 7.78T + 5.04e3T^{2} \) |
| 73 | \( 1 - 19.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 46.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 16.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 88.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 38.8iT - 9.40e3T^{2} \) |
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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
−10.76887633144245525727026030749, −9.352023376988527586875633695836, −9.049110496507760892316311358032, −8.219313198475211165543467839684, −7.16290331234943461367278981710, −5.83572940265188355820863294233, −5.09306204382835565709387437194, −3.54340088501640822184685471019, −1.94331921950573459286273299994, −0.49425205131555337700942275006,
1.60705834405515214079794036409, 3.06572317338829865076050093197, 4.18441402792074751195185737625, 5.97460736303094434447401661842, 6.75535779133174091044567656113, 7.58616052246807842196389573082, 8.464309045633190125395304855649, 9.686417926260351620752784685485, 10.54020971056607072411962035211, 10.83853601101067095652775401471
