| L(s) = 1 | + (−1.28 + 1.97i)2-s + (2.26 + 1.96i)3-s + (−0.624 − 1.40i)4-s + (4.99 − 0.228i)5-s + (−6.78 + 1.94i)6-s + (−0.592 − 0.158i)7-s + (−5.72 − 0.907i)8-s + (1.24 + 8.91i)9-s + (−5.94 + 10.1i)10-s + (−5.33 + 1.13i)11-s + (1.34 − 4.40i)12-s + (5.55 + 8.54i)13-s + (1.07 − 0.965i)14-s + (11.7 + 9.31i)15-s + (13.2 − 14.7i)16-s + (−3.00 + 18.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.640 + 0.986i)2-s + (0.754 + 0.656i)3-s + (−0.156 − 0.350i)4-s + (0.998 − 0.0456i)5-s + (−1.13 + 0.323i)6-s + (−0.0846 − 0.0226i)7-s + (−0.715 − 0.113i)8-s + (0.138 + 0.990i)9-s + (−0.594 + 1.01i)10-s + (−0.485 + 0.103i)11-s + (0.112 − 0.366i)12-s + (0.427 + 0.657i)13-s + (0.0765 − 0.0689i)14-s + (0.783 + 0.621i)15-s + (0.827 − 0.918i)16-s + (−0.176 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.379473 + 1.49976i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379473 + 1.49976i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.26 - 1.96i)T \) |
| 5 | \( 1 + (-4.99 + 0.228i)T \) |
| good | 2 | \( 1 + (1.28 - 1.97i)T + (-1.62 - 3.65i)T^{2} \) |
| 7 | \( 1 + (0.592 + 0.158i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (5.33 - 1.13i)T + (110. - 49.2i)T^{2} \) |
| 13 | \( 1 + (-5.55 - 8.54i)T + (-68.7 + 154. i)T^{2} \) |
| 17 | \( 1 + (3.00 - 18.9i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-5.36 - 7.38i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-0.132 + 2.53i)T + (-526. - 55.2i)T^{2} \) |
| 29 | \( 1 + (32.1 + 3.38i)T + (822. + 174. i)T^{2} \) |
| 31 | \( 1 + (4.34 + 41.3i)T + (-939. + 199. i)T^{2} \) |
| 37 | \( 1 + (-39.7 - 20.2i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (7.83 + 1.66i)T + (1.53e3 + 683. i)T^{2} \) |
| 43 | \( 1 + (-20.3 + 76.0i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (-27.7 - 34.3i)T + (-459. + 2.16e3i)T^{2} \) |
| 53 | \( 1 + (-2.23 - 14.1i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-16.0 + 75.5i)T + (-3.18e3 - 1.41e3i)T^{2} \) |
| 61 | \( 1 + (-33.3 + 7.08i)T + (3.39e3 - 1.51e3i)T^{2} \) |
| 67 | \( 1 + (-78.1 - 63.2i)T + (933. + 4.39e3i)T^{2} \) |
| 71 | \( 1 + (-4.34 - 3.15i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-21.3 + 10.8i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-117. - 12.3i)T + (6.10e3 + 1.29e3i)T^{2} \) |
| 83 | \( 1 + (11.1 + 29.1i)T + (-5.11e3 + 4.60e3i)T^{2} \) |
| 89 | \( 1 + (-65.3 - 21.2i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (38.7 + 47.8i)T + (-1.95e3 + 9.20e3i)T^{2} \) |
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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
−12.72904935983394106479812680058, −11.12446295625551123081976300260, −10.00139584559445647272315627443, −9.357456080710460353142033525222, −8.515093422664733110904655644504, −7.64874151241771341712589024719, −6.39538526720861399115883340480, −5.43196324090679838045135316686, −3.80390070678473209333733653009, −2.21249889726848344635672662388,
0.989160946682078893979308328665, 2.33392342468845034688525453859, 3.18551349576943539349916563857, 5.44729785192379228189404132576, 6.59491321893304565657654360301, 7.87528357122456425140515102401, 9.060605348466308879378850371054, 9.526072089412106271254714946078, 10.56053934073786517095952737886, 11.46803328053117609096458435648
