L(s) = 1 | + (0.839 − 0.224i)3-s + (−3.16 − 0.847i)5-s + (−0.654 + 2.56i)7-s + (−1.94 + 1.12i)9-s + (0.769 + 2.87i)11-s + (−3.63 − 3.63i)13-s − 2.84·15-s + (−1.81 + 3.14i)17-s + (0.429 − 1.60i)19-s + (0.0267 + 2.29i)21-s + (−5.33 + 3.08i)23-s + (4.95 + 2.86i)25-s + (−3.22 + 3.22i)27-s + (5.10 + 5.10i)29-s + (1.00 − 1.74i)31-s + ⋯ |
L(s) = 1 | + (0.484 − 0.129i)3-s + (−1.41 − 0.379i)5-s + (−0.247 + 0.968i)7-s + (−0.648 + 0.374i)9-s + (0.231 + 0.865i)11-s + (−1.00 − 1.00i)13-s − 0.734·15-s + (−0.441 + 0.763i)17-s + (0.0985 − 0.367i)19-s + (0.00584 + 0.501i)21-s + (−1.11 + 0.642i)23-s + (0.991 + 0.572i)25-s + (−0.620 + 0.620i)27-s + (0.947 + 0.947i)29-s + (0.180 − 0.313i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.144739 + 0.410834i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.144739 + 0.410834i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.654 - 2.56i)T \) |
good | 3 | \( 1 + (-0.839 + 0.224i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (3.16 + 0.847i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.769 - 2.87i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (3.63 + 3.63i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.81 - 3.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.429 + 1.60i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.33 - 3.08i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.10 - 5.10i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.00 + 1.74i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.57 + 1.49i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 3.71iT - 41T^{2} \) |
| 43 | \( 1 + (-2.91 + 2.91i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.06 - 8.77i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.986 + 3.68i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.977 + 3.64i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.75 + 6.54i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (5.88 - 1.57i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 9.55iT - 71T^{2} \) |
| 73 | \( 1 + (-0.989 - 0.571i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.120 + 0.209i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.459 - 0.459i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.76 - 2.17i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−11.67646226947407115124425506089, −10.60819182510101069909253109560, −9.466673499416545308815589873544, −8.538100304992382050625386226991, −7.969330225007650616021547590738, −7.11802912161448898512962515157, −5.65279465964820910457668057768, −4.65894577593175502364083436134, −3.43252598131689980381520593711, −2.28949872121977486681494584427,
0.24240657556836343975447511911, 2.77890356304014405047331060680, 3.79122335159419297744422381982, 4.50685854028333321910006727469, 6.29214303313033523470548100373, 7.16781696822106259965980582961, 7.993777517212544703295759212595, 8.806901128529058613078437467085, 9.855826140218220410041043966297, 10.81565519547085984594521697918