L(s) = 1 | + 9.55i·5-s + 9.27i·11-s − 4.24·13-s + 15.5i·17-s − 34.8·19-s − 17.7i·23-s − 66.3·25-s − 26.2i·29-s − 32.0·31-s + 55.3·37-s − 38.4i·41-s + 29.3·43-s + 68.2i·47-s − 1.08i·53-s − 88.6·55-s + ⋯ |
L(s) = 1 | + 1.91i·5-s + 0.843i·11-s − 0.326·13-s + 0.915i·17-s − 1.83·19-s − 0.772i·23-s − 2.65·25-s − 0.904i·29-s − 1.03·31-s + 1.49·37-s − 0.937i·41-s + 0.682·43-s + 1.45i·47-s − 0.0205i·53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4471655350\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4471655350\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 9.55iT - 25T^{2} \) |
| 11 | \( 1 - 9.27iT - 121T^{2} \) |
| 13 | \( 1 + 4.24T + 169T^{2} \) |
| 17 | \( 1 - 15.5iT - 289T^{2} \) |
| 19 | \( 1 + 34.8T + 361T^{2} \) |
| 23 | \( 1 + 17.7iT - 529T^{2} \) |
| 29 | \( 1 + 26.2iT - 841T^{2} \) |
| 31 | \( 1 + 32.0T + 961T^{2} \) |
| 37 | \( 1 - 55.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 29.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 68.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 1.08iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 47.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8.65T + 4.48e3T^{2} \) |
| 71 | \( 1 + 95.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 90.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 148.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 88.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 76.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 12.7T + 9.40e3T^{2} \) |
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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
−9.857328204334484458965987173512, −8.864064507149625755062680610226, −7.78002144953683897300984607748, −7.28776071364757120188160091971, −6.33482039325287121095720272998, −6.04277615644557054094177498811, −4.45263167181805015067251138923, −3.80865866240709194198520196079, −2.57446025297785277859077389674, −2.07535012365434887037994773936,
0.12140160820408946687575332703, 1.11363613402187220079407239634, 2.26337729490059930721476044829, 3.70703467345353664418581278477, 4.54157536877560183258290791832, 5.27634245068569499887123000388, 5.94348813631461434598721709763, 7.07661465550893096925849690848, 8.121851672062081422999710112073, 8.576397612200970721826347480737