| L(s) = 1 | + (−0.00423 + 0.00832i)2-s + (0.0529 + 1.73i)3-s + (1.17 + 1.61i)4-s + (0.362 − 2.28i)5-s + (−0.0146 − 0.00689i)6-s + (−1.04 + 1.22i)7-s + (−0.0368 + 0.00584i)8-s + (−2.99 + 0.183i)9-s + (0.0175 + 0.0127i)10-s + (2.45 + 3.99i)11-s + (−2.73 + 2.12i)12-s + (0.364 − 4.63i)13-s + (−0.00576 − 0.0139i)14-s + (3.98 + 0.506i)15-s + (−1.23 + 3.80i)16-s + (0.657 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.00299 + 0.00588i)2-s + (0.0305 + 0.999i)3-s + (0.587 + 0.808i)4-s + (0.162 − 1.02i)5-s + (−0.00597 − 0.00281i)6-s + (−0.396 + 0.463i)7-s + (−0.0130 + 0.00206i)8-s + (−0.998 + 0.0611i)9-s + (0.00553 + 0.00402i)10-s + (0.738 + 1.20i)11-s + (−0.790 + 0.612i)12-s + (0.101 − 1.28i)13-s + (−0.00154 − 0.00372i)14-s + (1.02 + 0.130i)15-s + (−0.308 + 0.950i)16-s + (0.159 − 0.664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01940 + 0.573180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01940 + 0.573180i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0529 - 1.73i)T \) |
| 41 | \( 1 + (6.40 + 0.141i)T \) |
| good | 2 | \( 1 + (0.00423 - 0.00832i)T + (-1.17 - 1.61i)T^{2} \) |
| 5 | \( 1 + (-0.362 + 2.28i)T + (-4.75 - 1.54i)T^{2} \) |
| 7 | \( 1 + (1.04 - 1.22i)T + (-1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-2.45 - 3.99i)T + (-4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (-0.364 + 4.63i)T + (-12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (-0.657 + 2.73i)T + (-15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (0.447 + 5.68i)T + (-18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (0.481 + 1.48i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.794 + 3.31i)T + (-25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (2.92 - 4.02i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.32 + 0.965i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (1.67 + 0.855i)T + (25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (-8.74 + 7.47i)T + (7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (-3.52 + 0.846i)T + (47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (8.07 - 2.62i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.62 - 7.11i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (4.76 - 7.77i)T + (-30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (13.1 - 8.04i)T + (32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (5.92 - 5.92i)T - 73iT^{2} \) |
| 79 | \( 1 + (-3.10 + 7.48i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 16.9iT - 83T^{2} \) |
| 89 | \( 1 + (-7.75 - 6.62i)T + (13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (11.4 + 6.99i)T + (44.0 + 86.4i)T^{2} \) |
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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
−13.36155792261421698403917632145, −12.44124664536703883653631534531, −11.71608109617161094103700729918, −10.40120460133787542693690239573, −9.253307642780581396893665840603, −8.559539521778825902659368223822, −7.10207311905210000255260392046, −5.49857132210962964158860518420, −4.30888502273105027274098339990, −2.79008676707123020933404110126,
1.69466434943571214589444482808, 3.41131361265235795306690315558, 6.04152056895029621009717579304, 6.44680167696686914738221322102, 7.49394822418537983033182904309, 9.037280620003752020999242201388, 10.39801480954526780606090013125, 11.21437375058465099464609112305, 12.04519144064086090182493150558, 13.56350342778749360462808702671
