| L(s) = 1 | + (0.719 + 1.21i)2-s + (1.36 + 1.06i)3-s + (−0.963 + 1.75i)4-s + (−1.63 + 1.54i)5-s + (−0.316 + 2.42i)6-s + (2.00 − 0.234i)7-s + (−2.82 + 0.0887i)8-s + (0.723 + 2.91i)9-s + (−3.05 − 0.879i)10-s + (−2.65 + 0.793i)11-s + (−3.18 + 1.36i)12-s + (4.43 − 0.258i)13-s + (1.73 + 2.27i)14-s + (−3.87 + 0.360i)15-s + (−2.14 − 3.37i)16-s + (−2.74 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (0.509 + 0.860i)2-s + (0.787 + 0.615i)3-s + (−0.481 + 0.876i)4-s + (−0.730 + 0.689i)5-s + (−0.129 + 0.991i)6-s + (0.759 − 0.0887i)7-s + (−0.999 + 0.0313i)8-s + (0.241 + 0.970i)9-s + (−0.965 − 0.278i)10-s + (−0.799 + 0.239i)11-s + (−0.919 + 0.393i)12-s + (1.22 − 0.0715i)13-s + (0.462 + 0.608i)14-s + (−1.00 + 0.0930i)15-s + (−0.535 − 0.844i)16-s + (−0.666 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.266099 + 2.06043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266099 + 2.06043i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.719 - 1.21i)T \) |
| 3 | \( 1 + (-1.36 - 1.06i)T \) |
| good | 5 | \( 1 + (1.63 - 1.54i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-2.00 + 0.234i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (2.65 - 0.793i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-4.43 + 0.258i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (2.74 + 1.00i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.362 - 0.994i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.722 - 0.0844i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (2.09 + 4.17i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-3.11 + 4.18i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-2.45 - 2.93i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-2.67 + 1.76i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (1.71 - 7.23i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-0.159 - 0.213i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-8.39 + 4.84i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.59 - 0.476i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-6.02 - 2.59i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (-4.23 + 8.43i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (-0.727 - 4.12i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.12 - 12.0i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-6.62 - 4.35i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-8.28 + 12.6i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (2.36 - 13.4i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-7.11 + 7.53i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−11.11358921186233572265112829725, −10.00154629006558517621143259009, −8.916450099169784139547253936189, −8.015693178485504748350485621347, −7.73654820859186951287828946815, −6.60406829831080335256555501408, −5.36769384297756671224678409496, −4.37647919987991430601934672418, −3.65181693219278432573606450331, −2.55303882534385982673486886781,
0.926499055046900820806254939021, 2.15425253031669390280314620114, 3.41732616156117892170024612808, 4.30801551469168344764696878782, 5.32260171128816916894304808163, 6.52196694542518359379635996992, 7.81321933811282060005130071839, 8.613880685119310912719211880712, 8.985216380292578231630613678215, 10.39696942901599545813142656428
