L(s) = 1 | + (1.97 + 1.13i)2-s + (1.59 + 2.75i)4-s + (0.717 + 1.24i)5-s + (−2.40 − 1.11i)7-s + 2.69i·8-s + 3.26i·10-s + (−2.80 − 1.61i)11-s + (4.43 − 2.55i)13-s + (−3.47 − 4.92i)14-s + (0.119 − 0.207i)16-s − 1.09·17-s + 4.48i·19-s + (−2.28 + 3.95i)20-s + (−3.68 − 6.37i)22-s + (−3.47 + 2.00i)23-s + ⋯ |
L(s) = 1 | + (1.39 + 0.804i)2-s + (0.795 + 1.37i)4-s + (0.320 + 0.555i)5-s + (−0.907 − 0.419i)7-s + 0.951i·8-s + 1.03i·10-s + (−0.844 − 0.487i)11-s + (1.22 − 0.709i)13-s + (−0.927 − 1.31i)14-s + (0.0298 − 0.0517i)16-s − 0.264·17-s + 1.02i·19-s + (−0.510 + 0.883i)20-s + (−0.784 − 1.35i)22-s + (−0.723 + 0.417i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87647 + 1.21081i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87647 + 1.21081i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.40 + 1.11i)T \) |
good | 2 | \( 1 + (-1.97 - 1.13i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.717 - 1.24i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.80 + 1.61i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.43 + 2.55i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.09T + 17T^{2} \) |
| 19 | \( 1 - 4.48iT - 19T^{2} \) |
| 23 | \( 1 + (3.47 - 2.00i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.02 + 0.593i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.24 - 1.87i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.239T + 37T^{2} \) |
| 41 | \( 1 + (3.71 + 6.43i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.11 + 3.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.01iT - 53T^{2} \) |
| 59 | \( 1 + (-4.73 - 8.20i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.82 + 1.63i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.82iT - 71T^{2} \) |
| 73 | \( 1 - 7.31iT - 73T^{2} \) |
| 79 | \( 1 + (1.83 - 3.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.45 + 9.44i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + (-2.69 - 1.55i)T + (48.5 + 84.0i)T^{2} \) |
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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
−13.15952723755064175762784244525, −12.22470399960516451313369601194, −10.81634194285049476698269939668, −10.02866191522700886308357888256, −8.333849557911435724737271464726, −7.22171907409034443890066274327, −6.17006687001346298839721624941, −5.60649272050962048466644119916, −3.92614880930788595164598737937, −3.04335907719612786680512741462,
2.08174355276751018544557511780, 3.42815671689783544475660044488, 4.68644442796253833564957911446, 5.69333478496586048373279816232, 6.71179010808577064339063086212, 8.575447166522664524252230288301, 9.628600098031014269397941764253, 10.76095303310469766563008151195, 11.64713067064845463435717163028, 12.68162318850598966435733523663