| L(s) = 1 | + (−0.381 − 2.65i)2-s + (0.654 − 0.755i)3-s + (−4.96 + 1.45i)4-s + (−2.85 + 1.83i)5-s + (−2.25 − 1.44i)6-s + (−0.558 − 3.88i)7-s + (3.52 + 7.71i)8-s + (−0.142 − 0.989i)9-s + (5.94 + 6.85i)10-s + (−2.19 + 1.41i)11-s + (−2.14 + 4.70i)12-s + (0.236 − 0.517i)13-s + (−10.0 + 2.96i)14-s + (−0.482 + 3.35i)15-s + (10.4 − 6.69i)16-s + (−2.66 − 0.782i)17-s + ⋯ |
| L(s) = 1 | + (−0.269 − 1.87i)2-s + (0.378 − 0.436i)3-s + (−2.48 + 0.728i)4-s + (−1.27 + 0.819i)5-s + (−0.919 − 0.590i)6-s + (−0.211 − 1.46i)7-s + (1.24 + 2.72i)8-s + (−0.0474 − 0.329i)9-s + (1.87 + 2.16i)10-s + (−0.662 + 0.425i)11-s + (−0.619 + 1.35i)12-s + (0.0655 − 0.143i)13-s + (−2.69 + 0.791i)14-s + (−0.124 + 0.866i)15-s + (2.60 − 1.67i)16-s + (−0.646 − 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.267481 + 0.391252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267481 + 0.391252i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 67 | \( 1 + (0.276 - 8.18i)T \) |
| good | 2 | \( 1 + (0.381 + 2.65i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (2.85 - 1.83i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.558 + 3.88i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (2.19 - 1.41i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.236 + 0.517i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (2.66 + 0.782i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.290 + 2.01i)T + (-18.2 - 5.35i)T^{2} \) |
| 23 | \( 1 + (-5.15 + 5.95i)T + (-3.27 - 22.7i)T^{2} \) |
| 29 | \( 1 + 9.79T + 29T^{2} \) |
| 31 | \( 1 + (2.94 + 6.44i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 - 0.00582T + 37T^{2} \) |
| 41 | \( 1 + (-5.68 - 1.66i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-4.23 - 1.24i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-0.540 + 0.624i)T + (-6.68 - 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.04 + 0.307i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (1.18 + 2.58i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.12 - 3.93i)T + (25.3 + 55.4i)T^{2} \) |
| 71 | \( 1 + (-9.44 + 2.77i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (7.50 + 4.82i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-0.827 + 1.81i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.200 - 0.128i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (6.76 + 7.80i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−11.39572317954703161989202845684, −11.01950342287683987845828109010, −10.20940760132422564463373219762, −9.047630722090038525164603080124, −7.78456086438963358607589004779, −7.17582170361307729048561636233, −4.44050050561157040451856469030, −3.65219010518999276783499188734, −2.54779165821176883264300178370, −0.43512895618167509769329167275,
3.73690768701602355905858774024, 5.05542904929384417500383426343, 5.71205655540414120279405257279, 7.29460902286078681153460469908, 8.139604385473322533223045136727, 8.895900313998392248652695225085, 9.342343984614970279564089212841, 11.13308112576216290409278725959, 12.51948077818112000330840713333, 13.26780830008682919376179516608
