L(s) = 1 | + (0.857 + 1.48i)2-s − 3-s + (−0.470 + 0.815i)4-s + (1.22 − 2.12i)5-s + (−0.857 − 1.48i)6-s + (2.18 − 1.49i)7-s + 1.81·8-s + 9-s + 4.21·10-s − 1.03·11-s + (0.470 − 0.815i)12-s + (−3.36 − 1.29i)13-s + (4.09 + 1.95i)14-s + (−1.22 + 2.12i)15-s + (2.49 + 4.32i)16-s + (−1.50 + 2.61i)17-s + ⋯ |
L(s) = 1 | + (0.606 + 1.05i)2-s − 0.577·3-s + (−0.235 + 0.407i)4-s + (0.549 − 0.951i)5-s + (−0.350 − 0.606i)6-s + (0.824 − 0.565i)7-s + 0.641·8-s + 0.333·9-s + 1.33·10-s − 0.313·11-s + (0.135 − 0.235i)12-s + (−0.932 − 0.360i)13-s + (1.09 + 0.523i)14-s + (−0.317 + 0.549i)15-s + (0.624 + 1.08i)16-s + (−0.366 + 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.66642 + 0.471233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.66642 + 0.471233i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + (-2.18 + 1.49i)T \) |
| 13 | \( 1 + (3.36 + 1.29i)T \) |
good | 2 | \( 1 + (-0.857 - 1.48i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 17 | \( 1 + (1.50 - 2.61i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + (-1.73 - 3.00i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.01 - 6.95i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.48 + 6.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.41 + 2.44i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.54 - 4.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.21 + 5.57i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.88 + 8.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.90 - 10.2i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.47 - 7.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 2.60T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + (1.63 + 2.82i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.50 - 13.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.211 - 0.366i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.34T + 83T^{2} \) |
| 89 | \( 1 + (2.65 + 4.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.92 - 5.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.22758448403397567826123932963, −11.01649916814851934144161700195, −10.19615906585404819626463282165, −8.958438258343625354837460042300, −7.69771458816490599056663491054, −7.06499350790696868290292066201, −5.50923821674895905144757338078, −5.29000492532547540328281400393, −4.19016429344109739918876632175, −1.53134382426272591091615554351,
1.98642670966409309136210609331, 2.93588623919755449457382933908, 4.58329449987000349139808927121, 5.40180406921861060258440844757, 6.78991853350869479604035013848, 7.75031506629496176858909592025, 9.390273648456909628436898106157, 10.34983303482888988767240937322, 11.06988109641720869103135329893, 11.75727791811565144327637818199