L(s) = 1 | − 1.08·5-s + (−2.10 + 1.60i)7-s + 1.23i·11-s − 3.69i·13-s − 6.30·17-s + 7.76i·19-s − 7.17i·23-s − 3.82·25-s − 1.41i·29-s − 4.54i·31-s + (2.27 − 1.74i)35-s − 2.82·37-s + 9.37·41-s + 7.68·43-s + 10.9·47-s + ⋯ |
L(s) = 1 | − 0.484·5-s + (−0.794 + 0.607i)7-s + 0.371i·11-s − 1.02i·13-s − 1.53·17-s + 1.78i·19-s − 1.49i·23-s − 0.765·25-s − 0.262i·29-s − 0.816i·31-s + (0.384 − 0.294i)35-s − 0.464·37-s + 1.46·41-s + 1.17·43-s + 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.060965114\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060965114\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.10 - 1.60i)T \) |
good | 5 | \( 1 + 1.08T + 5T^{2} \) |
| 11 | \( 1 - 1.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.69iT - 13T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 - 7.76iT - 19T^{2} \) |
| 23 | \( 1 + 7.17iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 4.54iT - 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 9.37T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.41iT - 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 61 | \( 1 - 9.55iT - 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 10.6iT - 71T^{2} \) |
| 73 | \( 1 - 4.59iT - 73T^{2} \) |
| 79 | \( 1 + 9.42T + 79T^{2} \) |
| 83 | \( 1 + 1.88T + 83T^{2} \) |
| 89 | \( 1 + 5.04T + 89T^{2} \) |
| 97 | \( 1 - 5.86iT - 97T^{2} \) |
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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
−8.374733094770543295383881070232, −7.74344233724581754809291656078, −6.97576675919501487160351646461, −5.96754843140118953687736651433, −5.77557472720273776906124894323, −4.38715502879830520158117405003, −3.94850795805984956822055182553, −2.80642176236956661924479706469, −2.14072186309762356823910429857, −0.48455761563029191750948343145,
0.65987932129543996026152924071, 2.08800306983987690405415543787, 3.07029924721182815273987490676, 3.99453389477135800055275602960, 4.47152060262589015670945260927, 5.53285898580179516969449831210, 6.48142771718493769925072473226, 7.03415255566785219333501234328, 7.51930140292655862335730400230, 8.645133552533689846624074666941