L(s) = 1 | + (1.15 − 1.99i)5-s + (2.41 + 1.08i)7-s + (−2.23 − 3.86i)11-s + (1.42 + 2.46i)13-s + (−0.115 + 0.199i)17-s + (−1.49 − 2.58i)19-s + (0.400 − 0.693i)23-s + (−0.149 − 0.259i)25-s + (3.82 − 6.62i)29-s + 5.28·31-s + (4.93 − 3.57i)35-s + (−1.69 − 2.93i)37-s + (−0.899 − 1.55i)41-s + (4.85 − 8.41i)43-s + 5.77·47-s + ⋯ |
L(s) = 1 | + (0.514 − 0.891i)5-s + (0.912 + 0.408i)7-s + (−0.672 − 1.16i)11-s + (0.394 + 0.682i)13-s + (−0.0279 + 0.0484i)17-s + (−0.342 − 0.593i)19-s + (0.0834 − 0.144i)23-s + (−0.0299 − 0.0518i)25-s + (0.710 − 1.23i)29-s + 0.949·31-s + (0.833 − 0.603i)35-s + (−0.278 − 0.482i)37-s + (−0.140 − 0.243i)41-s + (0.740 − 1.28i)43-s + 0.842·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.071575199\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.071575199\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.41 - 1.08i)T \) |
good | 5 | \( 1 + (-1.15 + 1.99i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.23 + 3.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.42 - 2.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.115 - 0.199i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.49 + 2.58i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.400 + 0.693i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.82 + 6.62i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.899 + 1.55i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.85 + 8.41i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 + (4.31 - 7.48i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.53T + 67T^{2} \) |
| 71 | \( 1 - 8.59T + 71T^{2} \) |
| 73 | \( 1 + (2.29 - 3.97i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 9.67T + 79T^{2} \) |
| 83 | \( 1 + (-8.46 + 14.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.944 - 1.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.70 + 13.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.885379102369917176945586709726, −8.278871778803755239780167108348, −7.53370796351261433678984457978, −6.27970123771231904547548315225, −5.72710280747216787467832281492, −4.89360213514029239437343418798, −4.24739520992627667373072403534, −2.87109905531757733934026257179, −1.90347070580121113275707236914, −0.76020088911155238777735589676,
1.34710922330629087411177284825, 2.38210824993391150391019219402, 3.25941036503148598365203163443, 4.50235670101194595597801424082, 5.09384891794232175356142517752, 6.15690748757366953593579491849, 6.82898359105846211426709720972, 7.73347774285970619430123800634, 8.137048239322418087562466670495, 9.249614193640690521021196331203