| 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
−10.63659543342498979412310840301, −9.646957480584113082706521261448, −8.300271369836555396844466913856, −8.211882223961005305678408012275, −6.69938686816711381382279741078, −6.35723634274131895621945361875, −4.34740176518170394951731285897, −3.57649891775045756903048866350, −2.26501247263839970299159662927, −0.15277321580865236805793190987,
1.03143563262433621000246767177, 3.50098225907357016498036928991, 4.54490432217730612217297223167, 5.58503880123386224242832116537, 6.89099280549671954851246802012, 7.34494174992257357550756926948, 8.314415627367750097554047007998, 9.104290940072439274041349960749, 10.08853932596099374314254690934, 11.05150660689633244194821198029
