L(s) = 1 | − 2-s + 4-s + (−0.111 + 0.192i)5-s + (−2.41 + 1.08i)7-s − 8-s + (0.111 − 0.192i)10-s + (2.19 − 3.80i)11-s + (2.36 − 2.71i)13-s + (2.41 − 1.08i)14-s + 16-s − 0.681·17-s + (0.376 + 0.651i)19-s + (−0.111 + 0.192i)20-s + (−2.19 + 3.80i)22-s − 6.63·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.0498 + 0.0862i)5-s + (−0.911 + 0.411i)7-s − 0.353·8-s + (0.0352 − 0.0610i)10-s + (0.662 − 1.14i)11-s + (0.656 − 0.754i)13-s + (0.644 − 0.290i)14-s + 0.250·16-s − 0.165·17-s + (0.0863 + 0.149i)19-s + (−0.0249 + 0.0431i)20-s + (−0.468 + 0.810i)22-s − 1.38·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5852650387\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5852650387\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.41 - 1.08i)T \) |
| 13 | \( 1 + (-2.36 + 2.71i)T \) |
good | 5 | \( 1 + (0.111 - 0.192i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.19 + 3.80i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 0.681T + 17T^{2} \) |
| 19 | \( 1 + (-0.376 - 0.651i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6.63T + 23T^{2} \) |
| 29 | \( 1 + (1.93 + 3.34i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.47 - 7.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.20T + 37T^{2} \) |
| 41 | \( 1 + (4.87 + 8.44i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.27 - 5.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.10 + 5.37i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.284 + 0.492i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.09T + 59T^{2} \) |
| 61 | \( 1 + (7.38 + 12.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.08 + 3.61i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.03 + 13.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.54 + 4.40i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.70 + 9.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.43T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + (-4.43 + 7.68i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.906886768097915332025492528363, −8.603942684091600497840021734760, −7.66817388781529859353460770430, −6.59160662670647458315817849891, −6.13991706185632109273452759151, −5.25503267696834163024388200054, −3.58432129304105190601366258037, −3.20316140901875840511060697667, −1.72975231033576425875561103263, −0.29435664042649745029043609957,
1.33847844917437188970415080465, 2.46961195356564502661590909043, 3.78102564556521911115761538317, 4.44107734048380280782087654689, 5.87891261642522776327605748032, 6.69169416269196489479041019150, 7.10146580295697423458065183675, 8.184381415189637921423818608342, 8.915190026598156028540651219674, 9.748820211345455776236475638441