L(s) = 1 | + (−1.63 + 1.14i)2-s + (1.37 − 3.75i)4-s + (−1.09 + 6.21i)5-s + (5.65 + 6.74i)7-s + (2.05 + 7.73i)8-s + (−5.32 − 11.4i)10-s + (3.98 − 0.702i)11-s + (11.3 − 4.11i)13-s + (−17.0 − 4.56i)14-s + (−12.2 − 10.3i)16-s + (7.63 − 13.2i)17-s + (−20.6 + 11.9i)19-s + (21.8 + 12.6i)20-s + (−5.72 + 5.71i)22-s + (−11.4 + 13.6i)23-s + ⋯ |
L(s) = 1 | + (−0.819 + 0.573i)2-s + (0.343 − 0.939i)4-s + (−0.218 + 1.24i)5-s + (0.808 + 0.963i)7-s + (0.256 + 0.966i)8-s + (−0.532 − 1.14i)10-s + (0.362 − 0.0638i)11-s + (0.870 − 0.316i)13-s + (−1.21 − 0.326i)14-s + (−0.764 − 0.644i)16-s + (0.449 − 0.777i)17-s + (−1.08 + 0.626i)19-s + (1.09 + 0.631i)20-s + (−0.260 + 0.259i)22-s + (−0.499 + 0.595i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.401410 + 0.984015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.401410 + 0.984015i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.63 - 1.14i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.09 - 6.21i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-5.65 - 6.74i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-3.98 + 0.702i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-11.3 + 4.11i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-7.63 + 13.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (20.6 - 11.9i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.4 - 13.6i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-40.6 - 14.8i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (31.8 - 38.0i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (14.4 - 24.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (40.7 - 14.8i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-19.2 + 3.39i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-2.31 - 2.75i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 81.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (48.7 + 8.58i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-79.8 + 67.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (23.6 + 64.8i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-48.4 - 27.9i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-12.9 - 22.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (28.9 - 79.6i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (21.4 - 58.8i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-11.6 - 20.1i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (16.4 + 93.0i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.37191848949249723146442075251, −10.80709665702375122075303037390, −9.887571329538588727237005560865, −8.677176203764794753483425966042, −8.096139731478360721339530013241, −6.94376255647268606063719985993, −6.16036997405369194260711196404, −5.07754075063494459492306552244, −3.18964303871423680828929877990, −1.72377787958668790436657812263,
0.69668269744801102934192337059, 1.80751318706560036154780577846, 3.91164156528373264189470702618, 4.54152325253996380595117957262, 6.32374090427133412141907606672, 7.61374197992080882901959573766, 8.433923730651276186585180138519, 8.964164857596102994610555040276, 10.23194774201421517096855852561, 10.95830448233162739675431027097