L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.72 + 0.158i)3-s + (−0.499 − 0.866i)4-s + (−0.724 − 1.25i)5-s + (−1 + 1.41i)6-s + (2.22 − 3.85i)7-s + 0.999·8-s + (2.94 + 0.548i)9-s + 1.44·10-s + (−0.5 + 0.866i)11-s + (−0.724 − 1.57i)12-s + (2.22 + 3.85i)13-s + (2.22 + 3.85i)14-s + (−1.05 − 2.28i)15-s + (−0.5 + 0.866i)16-s − 4.44·17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.995 + 0.0917i)3-s + (−0.249 − 0.433i)4-s + (−0.324 − 0.561i)5-s + (−0.408 + 0.577i)6-s + (0.840 − 1.45i)7-s + 0.353·8-s + (0.983 + 0.182i)9-s + 0.458·10-s + (−0.150 + 0.261i)11-s + (−0.209 − 0.454i)12-s + (0.617 + 1.06i)13-s + (0.594 + 1.02i)14-s + (−0.271 − 0.588i)15-s + (−0.125 + 0.216i)16-s − 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34119 + 0.111105i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34119 + 0.111105i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.72 - 0.158i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.724 + 1.25i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.22 + 3.85i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-2.22 - 3.85i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.67 - 4.63i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.55T + 37T^{2} \) |
| 41 | \( 1 + (1.67 + 2.89i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.224 - 0.389i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.94 + 3.37i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + (-0.275 - 0.476i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.89 - 11.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.17 - 3.76i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + (2.67 - 4.63i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4 + 6.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.79T + 89T^{2} \) |
| 97 | \( 1 + (5.39 - 9.35i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.93731066074545700242887584215, −11.23857805271348040380662966360, −10.45400710099286411943541459745, −9.155648018254670638613224164324, −8.499165044944569854238952675336, −7.53537519926816667846959414845, −6.74152623426404265738314088047, −4.64205554294573931131343200866, −4.06588353658832709183778938010, −1.61163280136181449351656857419,
2.13517902863921683897429506782, 3.09173343881339742077211369632, 4.62125682339698817433234934355, 6.33661082736559964162577674806, 7.901051339778262455761669577752, 8.498789073565105285742797559222, 9.243105331325957475351188871279, 10.69858713182660500721382144153, 11.24736495827911753848213594862, 12.57735020830069214619231176550