| L(s) = 1 | + (−2.40 + 0.511i)2-s + (1.05 − 1.37i)3-s + (3.70 − 1.65i)4-s + (1.50 − 1.65i)5-s + (−1.83 + 3.84i)6-s + (2.55 + 4.43i)7-s + (−4.09 + 2.97i)8-s + (−0.777 − 2.89i)9-s + (−2.77 + 4.75i)10-s + (1.54 − 0.329i)11-s + (1.63 − 6.83i)12-s + (−0.872 − 0.185i)13-s + (−8.42 − 9.36i)14-s + (−0.690 − 3.81i)15-s + (2.91 − 3.23i)16-s + (−3.04 + 2.21i)17-s + ⋯ |
| L(s) = 1 | + (−1.70 + 0.361i)2-s + (0.608 − 0.793i)3-s + (1.85 − 0.825i)4-s + (0.672 − 0.740i)5-s + (−0.748 + 1.57i)6-s + (0.967 + 1.67i)7-s + (−1.44 + 1.05i)8-s + (−0.259 − 0.965i)9-s + (−0.876 + 1.50i)10-s + (0.466 − 0.0992i)11-s + (0.473 − 1.97i)12-s + (−0.242 − 0.0514i)13-s + (−2.25 − 2.50i)14-s + (−0.178 − 0.983i)15-s + (0.727 − 0.808i)16-s + (−0.737 + 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.825647 - 0.153531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.825647 - 0.153531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.05 + 1.37i)T \) |
| 5 | \( 1 + (-1.50 + 1.65i)T \) |
| good | 2 | \( 1 + (2.40 - 0.511i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-2.55 - 4.43i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.54 + 0.329i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.872 + 0.185i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (3.04 - 2.21i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-3.69 + 2.68i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.66 - 4.07i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.00441 + 0.0419i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (0.205 + 1.95i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (1.43 + 4.43i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.748 + 0.159i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (4.11 + 7.13i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.426 - 4.05i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (2.00 + 1.45i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (9.31 + 1.98i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-7.08 + 1.50i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.375 + 3.57i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (1.57 + 1.14i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.42 - 13.6i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.355 - 3.38i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (3.54 + 1.57i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (0.935 - 2.87i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.544 + 5.17i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−11.95811606008138670655453164085, −11.22125316480446657979088666724, −9.592490272223338024301417054256, −8.974780941399707605815746443260, −8.554915561012391435116815991467, −7.57853291507278386895704485353, −6.37292084620221025815815464336, −5.34686444023841095777841413617, −2.37217661953907695930828195974, −1.45712282725750327559973617840,
1.61126849747766464269248337475, 3.11414980830254753453123270089, 4.67380285893843287411502785903, 6.85918210142016646607617591172, 7.56068971497328818437153077471, 8.558000792074767281761421331754, 9.605569752688581490782767560110, 10.22672453887119856266120925526, 10.86994037575764044255648398830, 11.52851520566800500125771564099
