| L(s) = 1 | + (−0.0909 + 1.03i)2-s + (1.49 − 0.874i)3-s + (0.896 + 0.158i)4-s + (−1.77 + 1.35i)5-s + (0.773 + 1.63i)6-s + (0.397 − 0.567i)7-s + (−0.786 + 2.93i)8-s + (1.46 − 2.61i)9-s + (−1.24 − 1.97i)10-s + (0.597 − 1.64i)11-s + (1.47 − 0.547i)12-s + (−0.720 + 0.0630i)13-s + (0.554 + 0.464i)14-s + (−1.46 + 3.58i)15-s + (−1.26 − 0.461i)16-s + (−0.792 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.0643 + 0.735i)2-s + (0.863 − 0.504i)3-s + (0.448 + 0.0790i)4-s + (−0.794 + 0.607i)5-s + (0.315 + 0.667i)6-s + (0.150 − 0.214i)7-s + (−0.277 + 1.03i)8-s + (0.489 − 0.871i)9-s + (−0.395 − 0.623i)10-s + (0.180 − 0.495i)11-s + (0.426 − 0.158i)12-s + (−0.199 + 0.0174i)13-s + (0.148 + 0.124i)14-s + (−0.379 + 0.925i)15-s + (−0.317 − 0.115i)16-s + (−0.192 − 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26633 + 0.504270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26633 + 0.504270i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.49 + 0.874i)T \) |
| 5 | \( 1 + (1.77 - 1.35i)T \) |
| good | 2 | \( 1 + (0.0909 - 1.03i)T + (-1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.397 + 0.567i)T + (-2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.597 + 1.64i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.720 - 0.0630i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (0.792 + 2.95i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.38 - 1.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.61 - 4.63i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-5.68 + 4.77i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.764 + 4.33i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (10.6 - 2.86i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.71 + 3.23i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.87 + 2.74i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (3.32 + 2.32i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-5.90 - 5.90i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.63 - 0.959i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 6.44i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.978 - 11.1i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-5.58 - 3.22i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.08 - 2.16i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.04 - 1.24i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.78 + 0.156i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (2.78 + 4.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.95 + 4.20i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−13.83696584668201126843219900266, −12.20790588119154421004889102968, −11.54755255316473796867442960739, −10.24457790054562576701836772612, −8.668221739758643279486782965193, −7.84299675749262642298347756940, −7.11719882407221072769354301588, −6.08684843078462622138991235834, −3.93135939687200933709893004471, −2.49157685689927125160868435091,
2.05841398813180675861836081882, 3.60175677679388307947443014245, 4.70960765197572784335709972004, 6.76067331859689417596083320508, 8.100660406700503861046400952647, 8.958111527777617215813715558460, 10.16890229305300846951236842028, 10.93909481209797469196895388920, 12.24737104237865632448133119434, 12.69088779909366720996540630100
