| L(s) = 1 | + (−1.18 + 0.777i)2-s + (−0.988 + 0.149i)3-s + (0.790 − 1.83i)4-s + (−3.50 − 1.37i)5-s + (1.05 − 0.944i)6-s + (−1.44 − 2.21i)7-s + (0.494 + 2.78i)8-s + (0.955 − 0.294i)9-s + (5.20 − 1.09i)10-s + (0.139 − 0.453i)11-s + (−0.508 + 1.93i)12-s + (2.84 − 0.649i)13-s + (3.42 + 1.49i)14-s + (3.66 + 0.837i)15-s + (−2.74 − 2.90i)16-s + (−4.01 − 5.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.835 + 0.549i)2-s + (−0.570 + 0.0860i)3-s + (0.395 − 0.918i)4-s + (−1.56 − 0.614i)5-s + (0.429 − 0.385i)6-s + (−0.545 − 0.837i)7-s + (0.174 + 0.984i)8-s + (0.318 − 0.0982i)9-s + (1.64 − 0.347i)10-s + (0.0421 − 0.136i)11-s + (−0.146 + 0.558i)12-s + (0.789 − 0.180i)13-s + (0.916 + 0.399i)14-s + (0.947 + 0.216i)15-s + (−0.687 − 0.726i)16-s + (−0.973 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0738979 + 0.134642i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0738979 + 0.134642i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.18 - 0.777i)T \) |
| 3 | \( 1 + (0.988 - 0.149i)T \) |
| 7 | \( 1 + (1.44 + 2.21i)T \) |
| good | 5 | \( 1 + (3.50 + 1.37i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.139 + 0.453i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-2.84 + 0.649i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (4.01 + 5.88i)T + (-6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-0.661 - 1.14i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.01 - 7.35i)T + (-8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-5.68 - 2.73i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (4.26 - 7.38i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.360 + 4.81i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 0.844i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.03 + 3.22i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (2.39 - 2.21i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.511 - 6.81i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-1.00 - 2.56i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-2.70 + 0.202i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-12.8 - 7.44i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.43 - 9.21i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (0.883 - 0.951i)T + (-5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (7.96 - 4.59i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.04 + 13.3i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (2.58 + 8.36i)T + (-73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 7.91iT - 97T^{2} \) |
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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
−11.05150660689633244194821198029, −10.08853932596099374314254690934, −9.104290940072439274041349960749, −8.314415627367750097554047007998, −7.34494174992257357550756926948, −6.89099280549671954851246802012, −5.58503880123386224242832116537, −4.54490432217730612217297223167, −3.50098225907357016498036928991, −1.03143563262433621000246767177,
0.15277321580865236805793190987, 2.26501247263839970299159662927, 3.57649891775045756903048866350, 4.34740176518170394951731285897, 6.35723634274131895621945361875, 6.69938686816711381382279741078, 8.211882223961005305678408012275, 8.300271369836555396844466913856, 9.646957480584113082706521261448, 10.63659543342498979412310840301
