L(s) = 1 | − 2.82·2-s − 5.19i·3-s + 8.00·4-s − 43.8i·5-s + 14.6i·6-s − 22.6·8-s − 27·9-s + 123. i·10-s + 66.2·11-s − 41.5i·12-s − 273. i·13-s − 227.·15-s + 64.0·16-s − 118. i·17-s + 76.3·18-s − 588. i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s − 1.75i·5-s + 0.408i·6-s − 0.353·8-s − 0.333·9-s + 1.23i·10-s + 0.547·11-s − 0.288i·12-s − 1.61i·13-s − 1.01·15-s + 0.250·16-s − 0.409i·17-s + 0.235·18-s − 1.63i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.182750410\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182750410\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82T \) |
| 3 | \( 1 + 5.19iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 43.8iT - 625T^{2} \) |
| 11 | \( 1 - 66.2T + 1.46e4T^{2} \) |
| 13 | \( 1 + 273. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 118. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 588. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 157.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.60e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 592. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 100.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.78e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 22.2T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.45e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 165.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.84e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 7.07e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.60e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.38e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.65e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 227.T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.15e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 370. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.20e4iT - 8.85e7T^{2} \) |
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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
−10.57047386399546491201234278632, −9.383008539379129028980691429070, −8.695148875849173026755288758226, −8.014977881224965730785246838796, −6.90071991057177308852788326523, −5.60684500571733468419133631757, −4.67160441109647762343515798254, −2.80941997527871219795346169104, −1.14629093394430805580474628204, −0.53387331261307044737256866574,
1.83987933684413202820093964641, 3.13838497265856133027044596014, 4.19664285228241632944072040260, 6.15330913310433080614590975124, 6.68139716155003345111255791678, 7.78409769983555358489945181358, 8.930717017697084424306321599233, 9.991003183627987068156663374730, 10.43528416688274974332908524385, 11.46963598498731009955076062972