| L(s) = 1 | + (−1.27 + 0.618i)2-s + (−1.15 + 1.28i)3-s + (1.23 − 1.57i)4-s + (1.21 + 0.651i)5-s + (0.672 − 2.35i)6-s + (1.31 + 0.260i)7-s + (−0.600 + 2.76i)8-s + (−0.327 − 2.98i)9-s + (−1.95 − 0.0753i)10-s + (1.62 − 1.33i)11-s + (0.599 + 3.41i)12-s + (4.29 − 2.29i)13-s + (−1.82 + 0.478i)14-s + (−2.24 + 0.819i)15-s + (−0.945 − 3.88i)16-s + (1.96 − 0.813i)17-s + ⋯ |
| L(s) = 1 | + (−0.899 + 0.437i)2-s + (−0.667 + 0.744i)3-s + (0.617 − 0.786i)4-s + (0.545 + 0.291i)5-s + (0.274 − 0.961i)6-s + (0.495 + 0.0985i)7-s + (−0.212 + 0.977i)8-s + (−0.109 − 0.994i)9-s + (−0.617 − 0.0238i)10-s + (0.489 − 0.402i)11-s + (0.173 + 0.984i)12-s + (1.19 − 0.636i)13-s + (−0.488 + 0.127i)14-s + (−0.580 + 0.211i)15-s + (−0.236 − 0.971i)16-s + (0.476 − 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.766007 + 0.475113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.766007 + 0.475113i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 - 0.618i)T \) |
| 3 | \( 1 + (1.15 - 1.28i)T \) |
| good | 5 | \( 1 + (-1.21 - 0.651i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (-1.31 - 0.260i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.62 + 1.33i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-4.29 + 2.29i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.96 + 0.813i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.802 - 2.64i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-2.52 - 1.68i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.95 - 1.60i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (3.06 - 3.06i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.30 - 7.59i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.25 - 2.17i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.725 - 7.36i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (6.67 - 2.76i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-7.57 + 6.21i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-7.16 - 3.83i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.633 + 6.43i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-5.99 + 0.590i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (14.9 + 2.97i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (1.48 + 7.46i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.48 + 8.42i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.84 + 9.38i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-4.91 - 7.35i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-8.47 - 8.47i)T + 97iT^{2} \) |
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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
−11.20829438748061405147651632418, −10.46343384818105861264200904816, −9.772861326382846375335891758397, −8.814994961690972784499614085564, −7.978653135827400256877453278657, −6.51109441395191525049800235540, −5.95892311954099276292659232607, −4.97377255896301744500426325711, −3.30253863740828671652276618281, −1.27459853647724199496592397952,
1.14489718192483942573499331100, 2.15881487904547500630776330580, 4.07103143908715051569016997362, 5.60184554473385247816663547677, 6.61974364299753124426131424668, 7.44831510357362874643898741822, 8.519104375921279031909748579521, 9.279008813136922486272990330916, 10.41935597499028216822845628682, 11.23308712802473027072759909762
