L(s) = 1 | − 7.74·5-s + 8.66·7-s − 13.4·11-s − 5.19i·13-s − 13.4i·17-s + 23i·19-s + 7.74i·23-s + 35.0·25-s − 30.9·29-s + 6.92·31-s − 67.0·35-s + 29.4i·37-s − 80.4i·41-s − 38i·43-s + 54.2i·47-s + ⋯ |
L(s) = 1 | − 1.54·5-s + 1.23·7-s − 1.21·11-s − 0.399i·13-s − 0.789i·17-s + 1.21i·19-s + 0.336i·23-s + 1.40·25-s − 1.06·29-s + 0.223·31-s − 1.91·35-s + 0.795i·37-s − 1.96i·41-s − 0.883i·43-s + 1.15i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.233757841\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233757841\) |
\(L(2)\) |
|
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 | \( 1 + 7.74T + 25T^{2} \) |
| 7 | \( 1 - 8.66T + 49T^{2} \) |
| 11 | \( 1 + 13.4T + 121T^{2} \) |
| 13 | \( 1 + 5.19iT - 169T^{2} \) |
| 17 | \( 1 + 13.4iT - 289T^{2} \) |
| 19 | \( 1 - 23iT - 361T^{2} \) |
| 23 | \( 1 - 7.74iT - 529T^{2} \) |
| 29 | \( 1 + 30.9T + 841T^{2} \) |
| 31 | \( 1 - 6.92T + 961T^{2} \) |
| 37 | \( 1 - 29.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 80.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 38iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 54.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 77.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 93.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 60.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 107iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 15.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 97T + 5.32e3T^{2} \) |
| 79 | \( 1 + 67.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 174. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 109T + 9.40e3T^{2} \) |
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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
−8.864990767593329089236354309896, −8.212494322064008038614447033412, −7.59402226340008508265275623504, −7.27589642615884421917369598574, −5.67146777424804045549918620058, −5.04827541727734967816126810481, −4.15815079387075288006147783767, −3.34384974649480329644939636349, −2.12637740653143087012634444035, −0.66200582221787305803118704566,
0.54754506055407886212326282791, 2.02950219550437015266969637470, 3.18203085704647042637715155335, 4.29352748311538407866457330800, 4.74189628440171381142691072087, 5.72370576168955038124687996115, 7.07282883899218452126391458241, 7.56261244080437173450469552700, 8.341419559013308286631296290826, 8.679011152392980342618985917172