| L(s) = 1 | + (−2 + 3.46i)2-s + (−8.58 + 13.0i)3-s + (−7.99 − 13.8i)4-s + (−39.2 − 67.9i)5-s + (−27.9 − 55.7i)6-s + (−110. + 191. i)7-s + 63.9·8-s + (−95.7 − 223. i)9-s + 313.·10-s + (−115. + 199. i)11-s + (248. + 14.7i)12-s + (385. + 668. i)13-s + (−442. − 766. i)14-s + (1.22e3 + 72.5i)15-s + (−128 + 221. i)16-s − 769.·17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.550 + 0.834i)3-s + (−0.249 − 0.433i)4-s + (−0.701 − 1.21i)5-s + (−0.316 − 0.632i)6-s + (−0.852 + 1.47i)7-s + 0.353·8-s + (−0.394 − 0.919i)9-s + 0.992·10-s + (−0.286 + 0.496i)11-s + (0.499 + 0.0296i)12-s + (0.633 + 1.09i)13-s + (−0.603 − 1.04i)14-s + (1.40 + 0.0832i)15-s + (−0.125 + 0.216i)16-s − 0.646·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0397338 - 0.343296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0397338 - 0.343296i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 3.46i)T \) |
| 3 | \( 1 + (8.58 - 13.0i)T \) |
| good | 5 | \( 1 + (39.2 + 67.9i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (110. - 191. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (115. - 199. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-385. - 668. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 769.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 383.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (193. + 334. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-394. + 683. i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.60e3 + 2.78e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 2.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-4.62e3 - 8.00e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (5.31e3 - 9.20e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (976. - 1.69e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.25e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.19e4 - 2.06e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.88e4 - 3.25e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.15e4 - 1.99e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-3.54e4 + 6.13e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.76e4 - 4.78e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 1.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-4.14e4 + 7.17e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−18.13779546372517718252372260423, −16.57129365209708683263168774578, −15.96915799238770267109285130681, −15.12463477000219797347209011207, −12.76934449698440983994196314707, −11.58187056062167237461055623628, −9.475077200999163593879622889209, −8.669307454575246483405434428531, −6.13191726385951281914000296910, −4.56636687833116194394944606844,
0.31637154417658173846555528120, 3.35908639154472099665748651899, 6.66979716421108013218873373971, 7.82908183812986805363425832654, 10.53172286475752414027897064341, 11.06252803263470322855786455432, 12.81795394538978867927819838011, 13.85946471812607869170124908440, 15.91606995175311338214245833678, 17.30555247170247858523707433008
