L(s) = 1 | − 1.17·2-s − 3-s − 0.623·4-s + (−2.20 + 0.344i)5-s + 1.17·6-s + (−1.71 − 2.01i)7-s + 3.07·8-s + 9-s + (2.59 − 0.404i)10-s + (1.48 − 2.96i)11-s + 0.623·12-s + 3.43i·13-s + (2.01 + 2.36i)14-s + (2.20 − 0.344i)15-s − 2.36·16-s + 3.10i·17-s + ⋯ |
L(s) = 1 | − 0.829·2-s − 0.577·3-s − 0.311·4-s + (−0.988 + 0.154i)5-s + 0.478·6-s + (−0.649 − 0.760i)7-s + 1.08·8-s + 0.333·9-s + (0.819 − 0.127i)10-s + (0.446 − 0.894i)11-s + 0.180·12-s + 0.952i·13-s + (0.538 + 0.631i)14-s + (0.570 − 0.0890i)15-s − 0.590·16-s + 0.752i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1666584857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1666584857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + (2.20 - 0.344i)T \) |
| 7 | \( 1 + (1.71 + 2.01i)T \) |
| 11 | \( 1 + (-1.48 + 2.96i)T \) |
good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 13 | \( 1 - 3.43iT - 13T^{2} \) |
| 17 | \( 1 - 3.10iT - 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 + 5.88iT - 23T^{2} \) |
| 29 | \( 1 - 5.81iT - 29T^{2} \) |
| 31 | \( 1 - 0.722iT - 31T^{2} \) |
| 37 | \( 1 + 1.11iT - 37T^{2} \) |
| 41 | \( 1 + 3.34T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 + 14.7iT - 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 + 1.98iT - 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 7.27iT - 73T^{2} \) |
| 79 | \( 1 + 11.5iT - 79T^{2} \) |
| 83 | \( 1 - 7.39iT - 83T^{2} \) |
| 89 | \( 1 - 3.65iT - 89T^{2} \) |
| 97 | \( 1 - 3.77T + 97T^{2} \) |
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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
−9.399220255133307569113982098823, −8.661369140579963330856647688619, −7.87203138221900308614555115633, −7.00347982140057127812366592070, −6.41757649294838956259610930703, −5.00757146125224832551926094938, −4.10424315756284756430554999044, −3.38943468809067670143918958686, −1.24840911329740307594920275739, −0.14199786459075084244285390182,
1.17173931543795422462426320061, 2.98598315914440824555292941002, 4.14575838963493387641302736454, 5.04304656460763581222039956904, 5.89764797946892677152586453852, 7.33645467269031137788546193714, 7.52042831146662646233448525350, 8.627541037670251615646251677925, 9.469391022506337884790036454650, 9.846115264395772591875592755471