L(s) = 1 | + (−5.41 + 1.63i)2-s + (4.17 − 15.0i)3-s + (26.6 − 17.7i)4-s − 73.1i·5-s + (1.97 + 88.1i)6-s + 51.8i·7-s + (−115. + 139. i)8-s + (−208. − 125. i)9-s + (119. + 395. i)10-s + 371.·11-s + (−155. − 474. i)12-s + 424.·13-s + (−84.8 − 280. i)14-s + (−1.09e3 − 305. i)15-s + (394. − 944. i)16-s + 1.40e3i·17-s + ⋯ |
L(s) = 1 | + (−0.957 + 0.289i)2-s + (0.268 − 0.963i)3-s + (0.832 − 0.554i)4-s − 1.30i·5-s + (0.0223 + 0.999i)6-s + 0.399i·7-s + (−0.636 + 0.771i)8-s + (−0.856 − 0.516i)9-s + (0.378 + 1.25i)10-s + 0.925·11-s + (−0.310 − 0.950i)12-s + 0.696·13-s + (−0.115 − 0.382i)14-s + (−1.26 − 0.350i)15-s + (0.385 − 0.922i)16-s + 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.720480 - 0.522350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.720480 - 0.522350i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.41 - 1.63i)T \) |
| 3 | \( 1 + (-4.17 + 15.0i)T \) |
good | 5 | \( 1 + 73.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 51.8iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 371.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 424.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.40e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 319. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.39e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.65e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 395.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.78e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.96e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.86e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.36e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.78e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.42e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.66e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 2.23e4T + 8.58e9T^{2} \) |
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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
−18.94111375871197164782807148163, −17.51950640024087280510455324199, −16.58598960240632682480926104060, −14.88002608156964868447259065866, −12.98933195158476555523958966029, −11.63230671926544909808404051002, −9.151777804060623101682587669533, −8.182448829984718223718398639380, −6.15609171820708310619378131278, −1.30326050094125470645358086998,
3.26438044507686794124104050459, 6.95991806357111451970752064714, 9.026773222648669868502847748085, 10.45512048170442275697797484455, 11.39565255385928655604159961759, 14.15837973774647300467059230406, 15.48324930718777922317530584738, 16.78778705005953742185620635009, 18.19139557471574660024624047436, 19.45199699851742493719750683332