L(s) = 1 | − 5.27i·7-s + (−4.46 + 4.46i)13-s + (2.44 − 2.44i)19-s − 5i·25-s − 4.52·31-s + (−8.46 − 8.46i)37-s + (7.34 + 7.34i)43-s − 20.8·49-s + (3.53 − 3.53i)61-s + (−11.3 + 11.3i)67-s + 13.8i·73-s − 15.0·79-s + (23.5 + 23.5i)91-s − 13.8·97-s − 16.5i·103-s + ⋯ |
L(s) = 1 | − 1.99i·7-s + (−1.23 + 1.23i)13-s + (0.561 − 0.561i)19-s − i·25-s − 0.811·31-s + (−1.39 − 1.39i)37-s + (1.12 + 1.12i)43-s − 2.97·49-s + (0.452 − 0.452i)61-s + (−1.38 + 1.38i)67-s + 1.62i·73-s − 1.69·79-s + (2.46 + 2.46i)91-s − 1.40·97-s − 1.63i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6231566339\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6231566339\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 5iT^{2} \) |
| 7 | \( 1 + 5.27iT - 7T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 13 | \( 1 + (4.46 - 4.46i)T - 13iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (-2.44 + 2.44i)T - 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29iT^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 + (8.46 + 8.46i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-7.34 - 7.34i)T + 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (-3.53 + 3.53i)T - 61iT^{2} \) |
| 67 | \( 1 + (11.3 - 11.3i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.670567457416820258281970242950, −7.52356047737426610021056374557, −7.22012075383567031562265034024, −6.59985720297621664020309615986, −5.34485602883561378180389492285, −4.36450233994296496395009454900, −4.01630549285453777700266110363, −2.74324972015038842783179014609, −1.49320818653316695854473762222, −0.20420525889157405788573879700,
1.74938485895175992015958681914, 2.72182985572843152092706780555, 3.38713189227330728940365645076, 4.94942738222165736245293428439, 5.42099782625177114540085559023, 6.00837330814840207211762155115, 7.17428220564439288606992079743, 7.88368588718839863404798708515, 8.688962548013182816343565096651, 9.287801577002026316216290263422