L(s) = 1 | + (−1.91 − 0.573i)2-s + (3.34 + 2.19i)4-s + (−2.18 + 3.78i)5-s + (1.84 − 1.06i)7-s + (−5.14 − 6.12i)8-s + (6.36 − 6.00i)10-s + (−10.9 + 6.30i)11-s + (−4.63 + 8.02i)13-s + (−4.15 + 0.984i)14-s + (6.33 + 14.6i)16-s + 14.6·17-s − 34.5i·19-s + (−15.6 + 7.85i)20-s + (24.5 − 5.81i)22-s + (−35.3 − 20.3i)23-s + ⋯ |
L(s) = 1 | + (−0.957 − 0.286i)2-s + (0.835 + 0.549i)4-s + (−0.437 + 0.757i)5-s + (0.264 − 0.152i)7-s + (−0.642 − 0.766i)8-s + (0.636 − 0.600i)10-s + (−0.993 + 0.573i)11-s + (−0.356 + 0.617i)13-s + (−0.296 + 0.0703i)14-s + (0.396 + 0.918i)16-s + 0.861·17-s − 1.81i·19-s + (−0.781 + 0.392i)20-s + (1.11 − 0.264i)22-s + (−1.53 − 0.886i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.000994364 - 0.0170748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.000994364 - 0.0170748i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.91 + 0.573i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.18 - 3.78i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.84 + 1.06i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (10.9 - 6.30i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (4.63 - 8.02i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 14.6T + 289T^{2} \) |
| 19 | \( 1 + 34.5iT - 361T^{2} \) |
| 23 | \( 1 + (35.3 + 20.3i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (9.52 + 16.4i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (0.860 + 0.496i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 66.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (12.9 - 22.4i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (36.4 - 21.0i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-30.1 + 17.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 12.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (49.1 + 28.3i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (36.8 + 63.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (82.4 + 47.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 75.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 56.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-64.5 + 37.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-56.7 + 32.7i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 150.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-56.4 - 97.7i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.67039099335590019737165723771, −10.78455240370048067707681711689, −10.15047952049403893560682548148, −9.164153966410456472685438570262, −7.978571116318691729721108721410, −7.35476514922645168819707101916, −6.45531461313527430619331660943, −4.75918954289174240881378131990, −3.24226506328250683745324420240, −2.10217957478837978210832754921,
0.01048425907924287681321207820, 1.66584185086688782797132152978, 3.41712546629884416637601065328, 5.24812765241880377255025155462, 5.87579596316802170646170991376, 7.56284303508558395949234498609, 8.041896044559294638691695050857, 8.801743666844785407492758616820, 10.11319939743872676915946989661, 10.51875804946675731868930095210