| L(s) = 1 | + (0.282 − 0.102i)2-s + (−1.46 + 1.22i)4-s + (0.173 + 0.984i)5-s + (−3.28 − 2.75i)7-s + (−0.588 + 1.01i)8-s + (0.150 + 0.260i)10-s + (−0.307 + 1.74i)11-s + (−3.80 − 1.38i)13-s + (−1.21 − 0.441i)14-s + (0.601 − 3.41i)16-s + (−1.95 − 3.38i)17-s + (−0.556 + 0.964i)19-s + (−1.46 − 1.22i)20-s + (0.0926 + 0.525i)22-s + (−6.60 + 5.54i)23-s + ⋯ |
| L(s) = 1 | + (0.199 − 0.0727i)2-s + (−0.731 + 0.613i)4-s + (0.0776 + 0.440i)5-s + (−1.24 − 1.04i)7-s + (−0.207 + 0.360i)8-s + (0.0475 + 0.0823i)10-s + (−0.0928 + 0.526i)11-s + (−1.05 − 0.383i)13-s + (−0.324 − 0.117i)14-s + (0.150 − 0.853i)16-s + (−0.473 − 0.820i)17-s + (−0.127 + 0.221i)19-s + (−0.327 − 0.274i)20-s + (0.0197 + 0.111i)22-s + (−1.37 + 1.15i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00375921 - 0.0655189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00375921 - 0.0655189i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| good | 2 | \( 1 + (-0.282 + 0.102i)T + (1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (3.28 + 2.75i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.307 - 1.74i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (3.80 + 1.38i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.556 - 0.964i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.60 - 5.54i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.562 + 0.204i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.24 + 1.04i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (2.60 + 4.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.711 - 0.258i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.25 - 7.10i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-9.00 - 7.55i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 3.97T + 53T^{2} \) |
| 59 | \( 1 + (-2.39 - 13.6i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.49 + 5.45i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (7.36 + 2.67i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.59 + 4.48i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.18 + 3.78i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-12.1 + 4.40i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.43 + 2.70i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (1.62 - 2.81i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.345 - 1.96i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.99188442535986485918644764475, −10.66498715100032850285506862312, −9.766848743044180040670175188352, −9.335792154610116560642185737342, −7.71423585182069430941712948056, −7.30027728360756537728273077251, −6.06081047311029344877199552531, −4.68406688457606722864202942012, −3.74202265376142046003912920081, −2.71610561014285295103052263113,
0.03796091769628332236622327631, 2.31614183049683612924244486681, 3.85306858792048839615280223660, 5.00335727189117129618165631381, 5.95456822432416473530626363804, 6.63516192382311151574778209628, 8.380214065873149432439204344663, 8.984064522693004472736109988882, 9.812767980280963325454774887060, 10.49783046370493270233900955782
