L(s) = 1 | + 1.66·5-s + 3.57i·7-s − 1.02·11-s + (1.87 − 3.07i)13-s − 5.05·17-s − 1.16·19-s − 8.17·23-s − 2.23·25-s − 4.29i·29-s − 7.98i·31-s + 5.93i·35-s − 9.83·37-s + 1.62i·41-s + 2.35i·43-s − 7.73i·47-s + ⋯ |
L(s) = 1 | + 0.743·5-s + 1.35i·7-s − 0.309·11-s + (0.520 − 0.853i)13-s − 1.22·17-s − 0.266·19-s − 1.70·23-s − 0.447·25-s − 0.798i·29-s − 1.43i·31-s + 1.00i·35-s − 1.61·37-s + 0.253i·41-s + 0.358i·43-s − 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3700331559\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3700331559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-1.87 + 3.07i)T \) |
good | 5 | \( 1 - 1.66T + 5T^{2} \) |
| 7 | \( 1 - 3.57iT - 7T^{2} \) |
| 11 | \( 1 + 1.02T + 11T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 + 4.29iT - 29T^{2} \) |
| 31 | \( 1 + 7.98iT - 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 - 1.62iT - 41T^{2} \) |
| 43 | \( 1 - 2.35iT - 43T^{2} \) |
| 47 | \( 1 + 7.73iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 9.73T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 10.5iT - 71T^{2} \) |
| 73 | \( 1 - 4.41iT - 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 + 9.02iT - 89T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−8.280279548606635937630868229264, −7.69750999823789497811765734004, −6.39387655987696423422136714144, −6.00752586986815843959739075153, −5.44261477504617260827658468315, −4.48325383532361876753940367855, −3.45803023276623709973206090105, −2.29695797841786120946815362154, −1.98913873278759726953898107865, −0.096095575243328341010274785497,
1.45893073111826272470711272288, 2.16610763414508762312846085095, 3.52210334031024839258262357697, 4.16827240197500294320290699406, 4.95713826757519131970446449965, 5.93368276828263308046740065215, 6.70173804448100577738530620851, 7.11050344898888606920162602156, 8.123408639553412245879740564778, 8.796658811288974348292446120600