| L(s) = 1 | − i·3-s − 3i·7-s − 9-s + 2·11-s − i·13-s − 2i·17-s + 5·19-s − 3·21-s − 6i·23-s + i·27-s + 10·29-s + 3·31-s − 2i·33-s + 2i·37-s − 39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.13i·7-s − 0.333·9-s + 0.603·11-s − 0.277i·13-s − 0.485i·17-s + 1.14·19-s − 0.654·21-s − 1.25i·23-s + 0.192i·27-s + 1.85·29-s + 0.538·31-s − 0.348i·33-s + 0.328i·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.983491005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.983491005\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−8.048787373294178350620862535016, −7.17308620956541651447560497629, −6.77931766684523473609719553630, −6.06771731384028623477623885545, −4.98901427664835785807480654895, −4.38681589508485589073896269040, −3.38041314016615848364295095934, −2.64447477526328708860233085579, −1.31374435277710617947574216176, −0.62701116250275862884048503316,
1.22143241512701422392711250804, 2.34344732904433365577925785050, 3.23311766744514547543812781387, 3.94990396447826469898531124589, 4.98652134563867613813000472289, 5.45887540204000471691637356250, 6.28882914090260720125246913109, 6.95669667609099563108475656357, 8.003492524450074869667556481498, 8.589124827348481471833685488340
