| L(s) = 1 | + (−0.540 + 1.30i)2-s + (0.634 + 0.773i)3-s + (−1.41 − 1.41i)4-s + (−3.76 − 1.14i)5-s + (−1.35 + 0.411i)6-s + (5.02 − 0.999i)7-s + (2.61 − 1.08i)8-s + (−0.195 + 0.980i)9-s + (3.52 − 4.30i)10-s + (4.06 − 0.399i)11-s + (0.193 − 1.99i)12-s + (−0.709 − 2.33i)13-s + (−1.40 + 7.10i)14-s + (−1.50 − 3.63i)15-s + (0.00918 + 3.99i)16-s + (−1.18 + 2.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.382 + 0.924i)2-s + (0.366 + 0.446i)3-s + (−0.707 − 0.706i)4-s + (−1.68 − 0.510i)5-s + (−0.552 + 0.167i)6-s + (1.89 − 0.377i)7-s + (0.923 − 0.384i)8-s + (−0.0650 + 0.326i)9-s + (1.11 − 1.36i)10-s + (1.22 − 0.120i)11-s + (0.0559 − 0.574i)12-s + (−0.196 − 0.648i)13-s + (−0.376 + 1.89i)14-s + (−0.388 − 0.938i)15-s + (0.00229 + 0.999i)16-s + (−0.286 + 0.692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04374 + 0.465171i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04374 + 0.465171i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.540 - 1.30i)T \) |
| 3 | \( 1 + (-0.634 - 0.773i)T \) |
| good | 5 | \( 1 + (3.76 + 1.14i)T + (4.15 + 2.77i)T^{2} \) |
| 7 | \( 1 + (-5.02 + 0.999i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-4.06 + 0.399i)T + (10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (0.709 + 2.33i)T + (-10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (1.18 - 2.85i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.81 - 0.971i)T + (10.5 + 15.7i)T^{2} \) |
| 23 | \( 1 + (-2.07 - 3.11i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.754 + 7.65i)T + (-28.4 - 5.65i)T^{2} \) |
| 31 | \( 1 + (-1.31 + 1.31i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.00 - 1.87i)T + (-20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (0.465 - 0.311i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-4.07 + 4.96i)T + (-8.38 - 42.1i)T^{2} \) |
| 47 | \( 1 + (-8.31 - 3.44i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.366 + 3.71i)T + (-51.9 + 10.3i)T^{2} \) |
| 59 | \( 1 + (-0.620 + 2.04i)T + (-49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (1.85 - 1.52i)T + (11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (10.0 - 8.28i)T + (13.0 - 65.7i)T^{2} \) |
| 71 | \( 1 + (1.13 + 5.69i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (3.80 + 0.756i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (1.75 - 0.728i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (1.95 - 3.66i)T + (-46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (7.24 - 10.8i)T + (-34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (7.28 - 7.28i)T - 97iT^{2} \) |
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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
−11.36853390837581799662986881824, −10.59503653351490203967415454480, −9.232505156587882021538969259559, −8.347363525253210630554151798424, −7.951079089047772495345202605659, −7.20720077854153634799368199680, −5.50122645685734783500442772544, −4.39055127222037418034901062683, −4.01306640929674143253082826830, −1.15190388911031242435958158626,
1.30282177444869246550711665212, 2.76414767779619233568605782918, 4.09063055059442864866586385865, 4.76463032935774223139549008511, 7.08145335300557495207057281422, 7.59593795035709521727355392407, 8.615160133296831348646438541501, 9.025745121113852212186900305087, 10.76067393621278940151790094657, 11.38501537180089423027500684328
