| L(s) = 1 | + (1.00 − 0.995i)2-s + (1.26 − 1.18i)3-s + (0.0190 − 1.99i)4-s + (0.476 − 1.57i)5-s + (0.0914 − 2.44i)6-s + (0.623 + 3.13i)7-s + (−1.97 − 2.02i)8-s + (0.195 − 2.99i)9-s + (−1.08 − 2.05i)10-s + (−1.36 + 0.134i)11-s + (−2.34 − 2.55i)12-s + (0.849 + 2.79i)13-s + (3.74 + 2.52i)14-s + (−1.25 − 2.54i)15-s + (−3.99 − 0.0761i)16-s + (1.79 + 0.744i)17-s + ⋯ |
| L(s) = 1 | + (0.710 − 0.703i)2-s + (0.729 − 0.683i)3-s + (0.00952 − 0.999i)4-s + (0.213 − 0.702i)5-s + (0.0373 − 0.999i)6-s + (0.235 + 1.18i)7-s + (−0.696 − 0.717i)8-s + (0.0651 − 0.997i)9-s + (−0.342 − 0.649i)10-s + (−0.410 + 0.0404i)11-s + (−0.676 − 0.736i)12-s + (0.235 + 0.776i)13-s + (1.00 + 0.675i)14-s + (−0.324 − 0.658i)15-s + (−0.999 − 0.0190i)16-s + (0.435 + 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44245 - 1.97950i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44245 - 1.97950i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.00 + 0.995i)T \) |
| 3 | \( 1 + (-1.26 + 1.18i)T \) |
| good | 5 | \( 1 + (-0.476 + 1.57i)T + (-4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (-0.623 - 3.13i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (1.36 - 0.134i)T + (10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-0.849 - 2.79i)T + (-10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (-1.79 - 0.744i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.623 - 1.16i)T + (-10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (-0.359 - 0.538i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (9.00 + 0.887i)T + (28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (-6.19 - 6.19i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.99 + 3.73i)T + (-20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (4.03 + 6.04i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (2.76 + 2.26i)T + (8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-10.7 - 4.46i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-7.82 + 0.770i)T + (51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (0.844 - 2.78i)T + (-49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-2.00 + 1.64i)T + (11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (4.94 + 6.02i)T + (-13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (-1.95 - 9.84i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (2.12 + 0.423i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (4.09 + 9.89i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (5.87 - 10.9i)T + (-46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (10.6 + 7.08i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (2.39 - 2.39i)T - 97iT^{2} \) |
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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
−11.43449628014708038856888901581, −10.14429094667688374907842926066, −9.002351238921320600469104504838, −8.722359903561586466999894614342, −7.24113357267988096560489102442, −5.98384154469861393545971474970, −5.19782233961863038552002620070, −3.81434165101712187746578772998, −2.50706557756075445615697400627, −1.52933025508700090019014482799,
2.69001835096544522523710981255, 3.64780961547905472591624563505, 4.61959644913737371495873455104, 5.73301170046325196472101015125, 7.05501026310711675813132130872, 7.73061955078079238115800500506, 8.579637206233003509855539316646, 9.912737951073255020289995802782, 10.61285212496139128824388457994, 11.47778845930087407926724564835
