L(s) = 1 | + (1.25 + 3.03i)3-s + (−0.824 + 1.98i)5-s + (0.0837 + 2.64i)7-s + (−5.52 + 5.52i)9-s + (0.623 + 0.258i)11-s + (−1.24 − 3.00i)13-s − 7.08·15-s + 2.68·17-s + (4.54 − 1.88i)19-s + (−7.92 + 3.58i)21-s + (0.871 + 0.871i)23-s + (0.256 + 0.256i)25-s + (−14.6 − 6.06i)27-s + (−2.60 − 6.29i)29-s + 4.75·31-s + ⋯ |
L(s) = 1 | + (0.726 + 1.75i)3-s + (−0.368 + 0.889i)5-s + (0.0316 + 0.999i)7-s + (−1.84 + 1.84i)9-s + (0.187 + 0.0778i)11-s + (−0.344 − 0.832i)13-s − 1.82·15-s + 0.650·17-s + (1.04 − 0.431i)19-s + (−1.73 + 0.781i)21-s + (0.181 + 0.181i)23-s + (0.0513 + 0.0513i)25-s + (−2.81 − 1.16i)27-s + (−0.483 − 1.16i)29-s + 0.853·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0351782 + 1.72512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0351782 + 1.72512i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.0837 - 2.64i)T \) |
good | 3 | \( 1 + (-1.25 - 3.03i)T + (-2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.824 - 1.98i)T + (-3.53 - 3.53i)T^{2} \) |
| 11 | \( 1 + (-0.623 - 0.258i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (1.24 + 3.00i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 2.68T + 17T^{2} \) |
| 19 | \( 1 + (-4.54 + 1.88i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.871 - 0.871i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.60 + 6.29i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 4.75T + 31T^{2} \) |
| 37 | \( 1 + (5.97 + 2.47i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-3.37 - 3.37i)T + 41iT^{2} \) |
| 43 | \( 1 + (-11.8 - 4.92i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 3.01iT - 47T^{2} \) |
| 53 | \( 1 + (-1.62 + 3.92i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (2.08 + 0.863i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-5.36 + 2.22i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (6.52 - 2.70i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (9.16 - 9.16i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.68 - 3.68i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.81T + 79T^{2} \) |
| 83 | \( 1 + (4.18 - 1.73i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.872 + 0.872i)T - 89iT^{2} \) |
| 97 | \( 1 + 16.0iT - 97T^{2} \) |
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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
−10.32536135965032690983782444883, −9.702548601342647365245976123007, −9.086189135009058738504426397708, −8.146967028002469093320820136052, −7.43245271099247924832852666038, −5.86758945002151462033760227304, −5.18752456460507389872549306935, −4.13338616940626576998728391656, −3.08439669521409364518419850117, −2.68003558278074296358196817064,
0.798617315533441978756994430879, 1.61568067808855747919650121843, 3.05761762830986786186581455460, 4.11192832649028131322013630001, 5.42794464571931604448054554615, 6.60889058823889535702207666237, 7.33568850910793623400891229319, 7.80623948004105603182689930412, 8.741660820899431478435468992444, 9.304278149438241094440818321180