L(s) = 1 | − 1.41·2-s + (−1.42 + 2.63i)3-s + 2.00·4-s + (−3.84 + 3.20i)5-s + (2.01 − 3.73i)6-s − 12.7i·7-s − 2.82·8-s + (−4.93 − 7.52i)9-s + (5.43 − 4.52i)10-s + 2.23i·11-s + (−2.85 + 5.27i)12-s + 0.635i·13-s + 17.9i·14-s + (−2.97 − 14.7i)15-s + 4.00·16-s − 9.66·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.475 + 0.879i)3-s + 0.500·4-s + (−0.768 + 0.640i)5-s + (0.336 − 0.622i)6-s − 1.81i·7-s − 0.353·8-s + (−0.547 − 0.836i)9-s + (0.543 − 0.452i)10-s + 0.203i·11-s + (−0.237 + 0.439i)12-s + 0.0489i·13-s + 1.28i·14-s + (−0.198 − 0.980i)15-s + 0.250·16-s − 0.568·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5687331141\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5687331141\) |
\(L(2)\) |
|
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 | \( 1 + (1.42 - 2.63i)T \) |
| 5 | \( 1 + (3.84 - 3.20i)T \) |
| 23 | \( 1 - 4.79T \) |
good | 7 | \( 1 + 12.7iT - 49T^{2} \) |
| 11 | \( 1 - 2.23iT - 121T^{2} \) |
| 13 | \( 1 - 0.635iT - 169T^{2} \) |
| 17 | \( 1 + 9.66T + 289T^{2} \) |
| 19 | \( 1 - 3.49T + 361T^{2} \) |
| 29 | \( 1 - 36.8iT - 841T^{2} \) |
| 31 | \( 1 - 38.8T + 961T^{2} \) |
| 37 | \( 1 - 2.05iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 20.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 15.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 103. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 66.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 62.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 21.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 12.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 110. iT - 9.40e3T^{2} \) |
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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
−10.41322814174145931712956088282, −10.00663131047476889249064314520, −8.875293135498050369980021121569, −7.84427887149401700068348823969, −7.03283196619079570683799558076, −6.41258785994171156760122535751, −4.79089490967689813169269044622, −3.99514312594382673797408289027, −3.07543924076071883203498336737, −0.898484413610325321496079907594,
0.37001809817623300701808697293, 1.85265777927169923824418464537, 2.91446972134936381492544189864, 4.76184720326959482210162612092, 5.72629292143172640700020280395, 6.47265425812263427746585796561, 7.61983249280022511278053831601, 8.402232115351204033038289627977, 8.819139825274532639602502859840, 9.871948990927874700023538972021