| L(s) = 1 | + (−1.23 − 0.685i)2-s + (2.29 + 0.397i)3-s + (1.06 + 1.69i)4-s + (−0.614 + 1.85i)5-s + (−2.56 − 2.06i)6-s + (−0.183 + 3.72i)7-s + (−0.149 − 2.82i)8-s + (2.27 + 0.814i)9-s + (2.03 − 1.87i)10-s + (−1.82 − 0.809i)11-s + (1.75 + 4.31i)12-s + (0.654 − 0.564i)13-s + (2.78 − 4.48i)14-s + (−2.14 + 4.01i)15-s + (−1.75 + 3.59i)16-s + (−0.995 + 0.532i)17-s + ⋯ |
| L(s) = 1 | + (−0.874 − 0.484i)2-s + (1.32 + 0.229i)3-s + (0.530 + 0.847i)4-s + (−0.274 + 0.831i)5-s + (−1.04 − 0.842i)6-s + (−0.0692 + 1.40i)7-s + (−0.0528 − 0.998i)8-s + (0.758 + 0.271i)9-s + (0.643 − 0.594i)10-s + (−0.550 − 0.244i)11-s + (0.507 + 1.24i)12-s + (0.181 − 0.156i)13-s + (0.743 − 1.19i)14-s + (−0.554 + 1.03i)15-s + (−0.437 + 0.899i)16-s + (−0.241 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03525 + 0.747618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03525 + 0.747618i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 0.685i)T \) |
| good | 3 | \( 1 + (-2.29 - 0.397i)T + (2.82 + 1.01i)T^{2} \) |
| 5 | \( 1 + (0.614 - 1.85i)T + (-4.01 - 2.97i)T^{2} \) |
| 7 | \( 1 + (0.183 - 3.72i)T + (-6.96 - 0.686i)T^{2} \) |
| 11 | \( 1 + (1.82 + 0.809i)T + (7.38 + 8.15i)T^{2} \) |
| 13 | \( 1 + (-0.654 + 0.564i)T + (1.90 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.995 - 0.532i)T + (9.44 - 14.1i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 1.31i)T + (9.76 - 16.2i)T^{2} \) |
| 23 | \( 1 + (-1.39 - 5.54i)T + (-20.2 + 10.8i)T^{2} \) |
| 29 | \( 1 + (5.30 - 0.130i)T + (28.9 - 1.42i)T^{2} \) |
| 31 | \( 1 + (3.16 - 2.11i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-10.2 - 1.26i)T + (35.8 + 8.99i)T^{2} \) |
| 41 | \( 1 + (1.00 - 0.744i)T + (11.9 - 39.2i)T^{2} \) |
| 43 | \( 1 + (1.92 - 2.72i)T + (-14.4 - 40.4i)T^{2} \) |
| 47 | \( 1 + (0.964 + 9.79i)T + (-46.0 + 9.16i)T^{2} \) |
| 53 | \( 1 + (-3.09 - 0.0760i)T + (52.9 + 2.60i)T^{2} \) |
| 59 | \( 1 + (-4.42 + 5.12i)T + (-8.65 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.34 - 0.302i)T + (55.1 - 26.0i)T^{2} \) |
| 67 | \( 1 + (-9.94 - 6.29i)T + (28.6 + 60.5i)T^{2} \) |
| 71 | \( 1 + (-14.8 - 7.03i)T + (45.0 + 54.8i)T^{2} \) |
| 73 | \( 1 + (0.209 + 4.26i)T + (-72.6 + 7.15i)T^{2} \) |
| 79 | \( 1 + (-7.59 - 9.25i)T + (-15.4 + 77.4i)T^{2} \) |
| 83 | \( 1 + (-2.01 + 0.248i)T + (80.5 - 20.1i)T^{2} \) |
| 89 | \( 1 + (-2.81 + 11.2i)T + (-78.4 - 41.9i)T^{2} \) |
| 97 | \( 1 + (-1.48 + 7.44i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−11.11919479530914136246292446923, −9.861449366731631170922293370696, −9.292543135346267907694052182695, −8.519917955267905755184728010422, −7.85257907702893379880213092433, −6.92385557001417305542572979909, −5.54336826514656660580490346470, −3.61448021849841994957142649033, −2.95156400009571755638466328984, −2.13436458538627192858071785265,
0.870354129707191570946587798608, 2.34289590122964640571636899129, 3.85513064076265032706646229070, 4.99840151929180728857198326672, 6.52907511324197756870222636897, 7.65892058800927925287544242610, 7.84097474044616305067591945321, 8.888638153586410436809893364549, 9.499535909826055566117870589633, 10.45349131228641346639392905677
