| L(s) = 1 | + (−1.21 + 1.59i)2-s + (1.49 + 2.60i)3-s + (−1.05 − 3.85i)4-s + (0.0372 + 0.211i)5-s + (−5.94 − 0.778i)6-s + (−7.43 + 8.86i)7-s + (7.41 + 2.99i)8-s + (−4.53 + 7.77i)9-s + (−0.381 − 0.197i)10-s + (9.50 + 1.67i)11-s + (8.45 − 8.51i)12-s + (−18.7 − 6.81i)13-s + (−5.08 − 22.5i)14-s + (−0.494 + 0.412i)15-s + (−13.7 + 8.17i)16-s + (−0.856 − 1.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.606 + 0.795i)2-s + (0.497 + 0.867i)3-s + (−0.264 − 0.964i)4-s + (0.00745 + 0.0422i)5-s + (−0.991 − 0.129i)6-s + (−1.06 + 1.26i)7-s + (0.927 + 0.373i)8-s + (−0.504 + 0.863i)9-s + (−0.0381 − 0.0197i)10-s + (0.864 + 0.152i)11-s + (0.704 − 0.709i)12-s + (−1.43 − 0.523i)13-s + (−0.362 − 1.61i)14-s + (−0.0329 + 0.0275i)15-s + (−0.859 + 0.510i)16-s + (−0.0503 − 0.0872i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.143017 + 0.883999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143017 + 0.883999i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.21 - 1.59i)T \) |
| 3 | \( 1 + (-1.49 - 2.60i)T \) |
| good | 5 | \( 1 + (-0.0372 - 0.211i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (7.43 - 8.86i)T + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-9.50 - 1.67i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (18.7 + 6.81i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (0.856 + 1.48i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-5.28 - 3.05i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-20.2 - 24.0i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-8.13 + 2.96i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-0.101 - 0.120i)T + (-166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-24.1 - 41.7i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-19.2 - 7.02i)T + (1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-45.0 - 7.94i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (23.9 - 28.4i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 98.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-12.4 + 2.18i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (62.8 + 52.7i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (18.1 - 49.9i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (65.5 - 37.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-13.3 + 23.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (39.9 + 109. i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (38.5 + 105. i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-59.4 + 102. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (4.27 - 24.2i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.48524924078300179152444575267, −13.07251152487263407970272518484, −11.73088246357780824339702567137, −10.19169258144236639811917454702, −9.443534776462809879084674693099, −8.827624333703716405470805255743, −7.40943651585532136548590208873, −6.01066810212399263572363594838, −4.84907285252669934991831068589, −2.82626907135879576209859635142,
0.77390704534760691570100111771, 2.73595760529952927811363674901, 4.10137471831861627567834200729, 6.83492530829704035174954606545, 7.32779844473555924092448162321, 8.889004868253637843535552387478, 9.650903170708369359340448078805, 10.83243819048685684294215569685, 12.14047891829408324898080323517, 12.81502352345527783963777469410
