L(s) = 1 | + (−1.94 − 2.68i)2-s + (0.927 + 2.85i)3-s + (−2.16 + 6.65i)4-s + (−3.89 + 5.36i)5-s + (5.84 − 8.04i)6-s + (−2.47 + 7.60i)7-s + (9.46 − 3.07i)8-s + (−7.28 + 5.29i)9-s + 22·10-s − 21.0·12-s + (−3.23 + 2.35i)13-s + (25.2 − 8.19i)14-s + (−18.9 − 6.14i)15-s + (−4.04 − 2.93i)16-s + (7.79 − 10.7i)17-s + (28.3 + 9.22i)18-s + ⋯ |
L(s) = 1 | + (−0.974 − 1.34i)2-s + (0.309 + 0.951i)3-s + (−0.540 + 1.66i)4-s + (−0.779 + 1.07i)5-s + (0.974 − 1.34i)6-s + (−0.353 + 1.08i)7-s + (1.18 − 0.384i)8-s + (−0.809 + 0.587i)9-s + 2.20·10-s − 1.75·12-s + (−0.248 + 0.180i)13-s + (1.80 − 0.585i)14-s + (−1.26 − 0.409i)15-s + (−0.252 − 0.183i)16-s + (0.458 − 0.631i)17-s + (1.57 + 0.512i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0110433 + 0.241166i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0110433 + 0.241166i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.927 - 2.85i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.94 + 2.68i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (3.89 - 5.36i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (2.47 - 7.60i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (3.23 - 2.35i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (-7.79 + 10.7i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (1.85 + 5.70i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 - 6.63iT - 529T^{2} \) |
| 29 | \( 1 + (37.8 + 12.2i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-21.0 + 15.2i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-9.27 + 28.5i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (12.6 - 4.09i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 - 42T + 1.84e3T^{2} \) |
| 47 | \( 1 + (82.0 - 26.6i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-35.0 - 48.2i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-63.0 - 20.4i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (9.70 + 7.05i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (35.0 - 48.2i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (22.8 - 70.3i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-32.3 + 23.5i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (23.3 - 32.1i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (50.1 - 36.4i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−11.44779610178983180713754086058, −10.76523559897147992484357595501, −9.771669208712052411515656793987, −9.301501279554762509864120792610, −8.330415164081494108915018493785, −7.39054731687890863778894381389, −5.73038008210167949597754912836, −4.08654882189245386765679876717, −3.07534440125568050306877978647, −2.44130470826850364898460714079,
0.16280659345968326138653496812, 1.20554694763034338443386311797, 3.69214226418847218809660842297, 5.17592101938620947801865004680, 6.38657828664411192061208095577, 7.20712448160143606610658286464, 7.975417168252991562310833272705, 8.441624861475450431688655351587, 9.423045473876552348156494938443, 10.42187396741473734994442495283