L(s) = 1 | + 5.35·2-s − 1.66i·3-s + 20.6·4-s + 5.96i·5-s − 8.89i·6-s − 27.8i·7-s + 67.6·8-s + 24.2·9-s + 31.9i·10-s + 18.6i·11-s − 34.3i·12-s − 42.5·13-s − 148. i·14-s + 9.92·15-s + 197.·16-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 0.319i·3-s + 2.58·4-s + 0.533i·5-s − 0.605i·6-s − 1.50i·7-s + 2.99·8-s + 0.897·9-s + 1.01i·10-s + 0.510i·11-s − 0.825i·12-s − 0.907·13-s − 2.84i·14-s + 0.170·15-s + 3.07·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(6.082949135\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.082949135\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
good | 2 | \( 1 - 5.35T + 8T^{2} \) |
| 3 | \( 1 + 1.66iT - 27T^{2} \) |
| 5 | \( 1 - 5.96iT - 125T^{2} \) |
| 7 | \( 1 + 27.8iT - 343T^{2} \) |
| 11 | \( 1 - 18.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 42.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 31.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 60.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 117. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 228. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 99.4iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 270. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 294.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 62.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 799. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 645.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.14e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 550. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 253. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 717.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 255. iT - 9.12e5T^{2} \) |
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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
−11.59434898500472526708432239363, −10.59153516113960763477038538467, −10.01911124353441603614195314082, −7.58480894181597684280607970305, −7.12468834812939722923441857571, −6.40468725210675726619979451519, −4.84751415585457325362701182020, −4.19715463414717376442211200495, −3.03896090414133088356855388100, −1.59241975525837217561504356150,
1.97756211765876270537495018452, 3.10095200152947351947591415955, 4.46027042360068683027180402557, 5.13863293338772288053554672899, 6.03762428741794652408179499464, 7.09587843649677502514314628737, 8.468086337971800717573082856995, 9.710227879755166665316460630353, 10.92295311122068026778565063060, 11.84407636680414696977592008119