| L(s) = 1 | + (1.37 + 1.45i)2-s + (−0.217 + 3.99i)4-s + (−1.64 − 2.84i)5-s + (−4.94 + 8.56i)7-s + (−6.09 + 5.17i)8-s + (1.87 − 6.30i)10-s + (−7.26 + 12.5i)11-s + (8.15 − 4.71i)13-s + (−19.2 + 4.59i)14-s + (−15.9 − 1.73i)16-s + 16.6i·17-s + 11.8i·19-s + (11.7 − 5.94i)20-s + (−28.2 + 6.75i)22-s + (27.6 − 15.9i)23-s + ⋯ |
| L(s) = 1 | + (0.687 + 0.726i)2-s + (−0.0542 + 0.998i)4-s + (−0.328 − 0.569i)5-s + (−0.706 + 1.22i)7-s + (−0.762 + 0.647i)8-s + (0.187 − 0.630i)10-s + (−0.660 + 1.14i)11-s + (0.627 − 0.362i)13-s + (−1.37 + 0.328i)14-s + (−0.994 − 0.108i)16-s + 0.977i·17-s + 0.622i·19-s + (0.586 − 0.297i)20-s + (−1.28 + 0.306i)22-s + (1.20 − 0.693i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.442753 + 1.49441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442753 + 1.49441i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 - 1.45i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.64 + 2.84i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (4.94 - 8.56i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (7.26 - 12.5i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-8.15 + 4.71i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 16.6iT - 289T^{2} \) |
| 19 | \( 1 - 11.8iT - 361T^{2} \) |
| 23 | \( 1 + (-27.6 + 15.9i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (13.0 - 22.6i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (8.69 + 15.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 40.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-6.97 + 4.02i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-50.5 - 29.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (0.0690 + 0.0398i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 10.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-53.8 - 93.3i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-46.2 - 26.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (69.3 - 40.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 16.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 30.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-46.7 + 81.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (7.74 - 13.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 12.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (91.1 - 157. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.81476926140312931777528674565, −12.03317226665449546749183416385, −10.62616474287133287597965760970, −9.165381605878503116971699653426, −8.468978053287334766301223458396, −7.37525325600023060583982406002, −6.12351559118661000229633666501, −5.28590676739186421351825301460, −4.05082655964633775483810941362, −2.58965868653525226257243993946,
0.69480533806778147481036897275, 2.98158154586038137190634219428, 3.72886378896303077303712610883, 5.16111520293919730110928552235, 6.50042825308672420731558184692, 7.35647569338908214859631174620, 9.014070138566739860195617834655, 10.06025731199779470764045765200, 11.12848961675152852252731883496, 11.27445004986481384865901873606
