L(s) = 1 | − 0.692·3-s + (−0.245 + 2.22i)5-s + (0.343 + 0.343i)7-s − 2.52·9-s + (−0.843 + 0.843i)11-s + 3.68i·13-s + (0.169 − 1.53i)15-s + (0.412 + 0.412i)17-s + (−5.37 + 5.37i)19-s + (−0.238 − 0.238i)21-s + (3.08 − 3.08i)23-s + (−4.87 − 1.09i)25-s + 3.82·27-s + (4.22 + 4.22i)29-s + 8.75i·31-s + ⋯ |
L(s) = 1 | − 0.399·3-s + (−0.109 + 0.993i)5-s + (0.129 + 0.129i)7-s − 0.840·9-s + (−0.254 + 0.254i)11-s + 1.02i·13-s + (0.0438 − 0.397i)15-s + (0.0999 + 0.0999i)17-s + (−1.23 + 1.23i)19-s + (−0.0519 − 0.0519i)21-s + (0.643 − 0.643i)23-s + (−0.975 − 0.218i)25-s + 0.735·27-s + (0.785 + 0.785i)29-s + 1.57i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.465864 + 0.680586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.465864 + 0.680586i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.245 - 2.22i)T \) |
good | 3 | \( 1 + 0.692T + 3T^{2} \) |
| 7 | \( 1 + (-0.343 - 0.343i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.843 - 0.843i)T - 11iT^{2} \) |
| 13 | \( 1 - 3.68iT - 13T^{2} \) |
| 17 | \( 1 + (-0.412 - 0.412i)T + 17iT^{2} \) |
| 19 | \( 1 + (5.37 - 5.37i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3.08 + 3.08i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.22 - 4.22i)T + 29iT^{2} \) |
| 31 | \( 1 - 8.75iT - 31T^{2} \) |
| 37 | \( 1 + 5.41iT - 37T^{2} \) |
| 41 | \( 1 + 2.54iT - 41T^{2} \) |
| 43 | \( 1 - 4.30iT - 43T^{2} \) |
| 47 | \( 1 + (-4.56 + 4.56i)T - 47iT^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + (7.33 + 7.33i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.81 - 4.81i)T - 61iT^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 - 2.97T + 71T^{2} \) |
| 73 | \( 1 + (-6.87 - 6.87i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 7.15T + 83T^{2} \) |
| 89 | \( 1 + 1.10T + 89T^{2} \) |
| 97 | \( 1 + (-7.15 - 7.15i)T + 97iT^{2} \) |
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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
−11.91677223851451678470981920499, −10.79560728503505961638383358566, −10.48791845115481908938124940481, −9.054250331996990458814088413439, −8.156397878404055624034841118029, −6.88174340177603819553784099387, −6.22905281329739194752933001377, −4.98252592051903412828630919381, −3.58954923358416996914856367121, −2.22664393660436859106675981715,
0.60705484405449723320605318930, 2.74051281981871487175196904781, 4.39030650150776445955721284104, 5.35042616170829928262152811313, 6.22055935928601732766877996474, 7.72157550182139914889215359668, 8.495391871920910986702709153546, 9.362872763616423464997435730152, 10.60053941763559023998102420701, 11.36269718706074739676570157661