| L(s) = 1 | + 11.2i·2-s − 11.4i·3-s − 93.8·4-s + 128.·6-s − 132. i·7-s − 693. i·8-s + 111.·9-s + 121·11-s + 1.07e3i·12-s + 916. i·13-s + 1.48e3·14-s + 4.77e3·16-s − 607. i·17-s + 1.24e3i·18-s − 2.34e3·19-s + ⋯ |
| L(s) = 1 | + 1.98i·2-s − 0.735i·3-s − 2.93·4-s + 1.45·6-s − 1.02i·7-s − 3.83i·8-s + 0.458·9-s + 0.301·11-s + 2.15i·12-s + 1.50i·13-s + 2.02·14-s + 4.66·16-s − 0.509i·17-s + 0.909i·18-s − 1.48·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2582867030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2582867030\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 11.2iT - 32T^{2} \) |
| 3 | \( 1 + 11.4iT - 243T^{2} \) |
| 7 | \( 1 + 132. iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 916. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 607. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.34e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 15.5iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 837.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.91e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 9.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.47e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.63e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.05e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.83e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.07e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.80e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.39e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.36e4iT - 8.58e9T^{2} \) |
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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
−12.05207168955133215054634971574, −10.34984046666829692837260595497, −9.366935466016718350138722468881, −8.431176459903252888412180405680, −7.44336491258240615177853824948, −6.81031820540653850368027623645, −6.27528348467192883487898472112, −4.63920819300251500404086946945, −4.06057865637959780359717371793, −1.27695965252926868231924288914,
0.079393949221094661740183431701, 1.63304860749762783284008692106, 2.78793500725743495945205102634, 3.80706530551625073176643717043, 4.77439747235357444397247874616, 5.77637955080112413864842220819, 8.162741207769414334031600219763, 8.907387425027373999062390294507, 9.783824193041299157563436446837, 10.52664334803501831572786390113
