L(s) = 1 | + (−1.11 + 0.870i)2-s + (−0.755 + 0.654i)3-s + (0.484 − 1.94i)4-s + (0.714 − 0.102i)5-s + (0.272 − 1.38i)6-s + (−0.269 + 0.589i)7-s + (1.14 + 2.58i)8-s + (0.142 − 0.989i)9-s + (−0.707 + 0.736i)10-s + (1.63 − 5.56i)11-s + (0.904 + 1.78i)12-s + (−2.71 + 1.24i)13-s + (−0.213 − 0.890i)14-s + (−0.472 + 0.545i)15-s + (−3.53 − 1.87i)16-s + (1.82 + 1.17i)17-s + ⋯ |
L(s) = 1 | + (−0.788 + 0.615i)2-s + (−0.436 + 0.378i)3-s + (0.242 − 0.970i)4-s + (0.319 − 0.0459i)5-s + (0.111 − 0.566i)6-s + (−0.101 + 0.222i)7-s + (0.406 + 0.913i)8-s + (0.0474 − 0.329i)9-s + (−0.223 + 0.233i)10-s + (0.492 − 1.67i)11-s + (0.261 + 0.514i)12-s + (−0.753 + 0.344i)13-s + (−0.0569 − 0.238i)14-s + (−0.122 + 0.140i)15-s + (−0.882 − 0.469i)16-s + (0.443 + 0.284i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.775587 - 0.123770i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.775587 - 0.123770i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.11 - 0.870i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (-4.59 + 1.36i)T \) |
good | 5 | \( 1 + (-0.714 + 0.102i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.269 - 0.589i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.63 + 5.56i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.71 - 1.24i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.82 - 1.17i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.21 + 5.00i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.15 + 4.90i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.19 + 5.99i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (3.87 + 0.557i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.0678 + 0.471i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-2.94 + 2.55i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 + (-4.06 - 1.85i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.0302 - 0.0138i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-5.24 - 4.54i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (2.91 + 9.92i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.32 + 0.682i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-7.15 + 4.59i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.81 - 12.7i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (9.17 + 1.31i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (10.0 + 11.5i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.111 + 0.774i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.67678855386883516685462513849, −9.687782809334195560605303876382, −9.016492351379477002337632632970, −8.264009816664461845049603854280, −7.04060388509235585766536344129, −6.14526818576044662221574183920, −5.52842420691962585898718003426, −4.30097833357434287603561511232, −2.56829425245690891948895619812, −0.67670993023436268740400389158,
1.37125211850148723319737823902, 2.50841589249072661527287727725, 4.01485749301277668915072911851, 5.18127510113390934698538219468, 6.68989940348747624495177024742, 7.22470190142230071223529834322, 8.174593381405679859476124317355, 9.305927716696261495034708321442, 10.13878697301449663391324051757, 10.50102708862520622459706426834