| L(s) = 1 | + (−1.08 + 1.67i)2-s + (−2.24 − 1.98i)3-s + (−1.62 − 3.65i)4-s + (0.641 + 3.64i)5-s + (5.78 − 1.59i)6-s + (2.33 − 2.78i)7-s + (7.90 + 1.25i)8-s + (1.08 + 8.93i)9-s + (−6.80 − 2.89i)10-s + (15.6 + 2.76i)11-s + (−3.61 + 11.4i)12-s + (13.1 + 4.78i)13-s + (2.12 + 6.95i)14-s + (5.80 − 9.45i)15-s + (−10.7 + 11.8i)16-s + (4.98 + 8.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.544 + 0.838i)2-s + (−0.748 − 0.663i)3-s + (−0.406 − 0.913i)4-s + (0.128 + 0.728i)5-s + (0.963 − 0.266i)6-s + (0.333 − 0.397i)7-s + (0.987 + 0.156i)8-s + (0.120 + 0.992i)9-s + (−0.680 − 0.289i)10-s + (1.42 + 0.251i)11-s + (−0.301 + 0.953i)12-s + (1.01 + 0.367i)13-s + (0.151 + 0.496i)14-s + (0.386 − 0.630i)15-s + (−0.669 + 0.742i)16-s + (0.293 + 0.507i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.826811 + 0.351292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.826811 + 0.351292i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.08 - 1.67i)T \) |
| 3 | \( 1 + (2.24 + 1.98i)T \) |
| good | 5 | \( 1 + (-0.641 - 3.64i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.33 + 2.78i)T + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-15.6 - 2.76i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-13.1 - 4.78i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-4.98 - 8.63i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (14.6 + 8.46i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.374 - 0.446i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-16.6 + 6.04i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-18.3 - 21.8i)T + (-166. + 946. i)T^{2} \) |
| 37 | \( 1 + (31.7 + 55.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-59.2 - 21.5i)T + (1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (1.70 + 0.300i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (51.7 - 61.6i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 66.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (65.4 - 11.5i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-5.43 - 4.55i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-12.7 + 35.1i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (12.7 - 7.38i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-66.1 + 114. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-24.4 - 67.1i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (8.25 + 22.6i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (29.6 - 51.3i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-13.3 + 75.5i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.98818034371915661102986598208, −12.59949476391395978918943675060, −11.16221210987877335425727314773, −10.61128956140273515171941324585, −9.124685039947313160393492834528, −7.86331396981996919093123801426, −6.65314811099593149219009330194, −6.23003329943980467041868442281, −4.45522374898636021516215056084, −1.39103163863817116340183433526,
1.14405162026324001025051364040, 3.65816587940860386411814334296, 4.88897697883979866518958526694, 6.40030163781469979867905834602, 8.422197444643829188678146783709, 9.124684689193667100505272689573, 10.19389454006360981701503902576, 11.31625897203040899239265384791, 11.95628394776696899926740250926, 12.90734794260554765338885840051
