L(s) = 1 | + (−1.15 − 0.819i)2-s + (2.04 + 2.04i)3-s + (0.655 + 1.88i)4-s + (0.814 − 0.814i)5-s + (−0.680 − 4.03i)6-s − 2.42i·7-s + (0.794 − 2.71i)8-s + 5.38i·9-s + (−1.60 + 0.270i)10-s + (−0.707 + 0.707i)11-s + (−2.52 + 5.21i)12-s + (4.19 + 4.19i)13-s + (−1.98 + 2.79i)14-s + 3.33·15-s + (−3.14 + 2.47i)16-s − 3.70·17-s + ⋯ |
L(s) = 1 | + (−0.814 − 0.579i)2-s + (1.18 + 1.18i)3-s + (0.327 + 0.944i)4-s + (0.364 − 0.364i)5-s + (−0.277 − 1.64i)6-s − 0.916i·7-s + (0.280 − 0.959i)8-s + 1.79i·9-s + (−0.508 + 0.0856i)10-s + (−0.213 + 0.213i)11-s + (−0.729 + 1.50i)12-s + (1.16 + 1.16i)13-s + (−0.531 + 0.746i)14-s + 0.861·15-s + (−0.785 + 0.619i)16-s − 0.897·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17683 + 0.162845i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17683 + 0.162845i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.15 + 0.819i)T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-2.04 - 2.04i)T + 3iT^{2} \) |
| 5 | \( 1 + (-0.814 + 0.814i)T - 5iT^{2} \) |
| 7 | \( 1 + 2.42iT - 7T^{2} \) |
| 13 | \( 1 + (-4.19 - 4.19i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 19 | \( 1 + (1.91 + 1.91i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.56iT - 23T^{2} \) |
| 29 | \( 1 + (2.07 + 2.07i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 + (-5.40 + 5.40i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.68iT - 41T^{2} \) |
| 43 | \( 1 + (4.92 - 4.92i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.29T + 47T^{2} \) |
| 53 | \( 1 + (1.72 - 1.72i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.02 + 8.02i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.82 + 5.82i)T + 61iT^{2} \) |
| 67 | \( 1 + (5.59 + 5.59i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.42iT - 71T^{2} \) |
| 73 | \( 1 - 16.1iT - 73T^{2} \) |
| 79 | \( 1 - 5.56T + 79T^{2} \) |
| 83 | \( 1 + (1.72 + 1.72i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.5iT - 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−12.97676075593630706017791200359, −11.16758543284843849600116880914, −10.65908911281381507404730393493, −9.514474377678400264101407590301, −9.021154439873771601780343785782, −8.148621503513454181717597114788, −6.80877787047934647394704447707, −4.45757414720440223529544503803, −3.70808041666335096041899082437, −2.10101360936661096144344541944,
1.72697433902593104414351060292, 2.97702609820403254570224821552, 5.74475702848951376418747157395, 6.54903079200240760987344201677, 7.76733561480333427416789921137, 8.456894456264853182987840103426, 9.170230490297137136463662494439, 10.43809476510334605405314854329, 11.64415358972326828448762195537, 13.08191051128765430948272357601