| L(s) = 1 | + (1.73 + 7.61i)2-s + (8.10 + 3.90i)3-s + (−26.1 + 12.5i)4-s + (−21.6 − 94.9i)5-s + (−15.6 + 68.5i)6-s + (−141. − 68.1i)7-s + (14.5 + 18.2i)8-s + (50.5 + 63.3i)9-s + (685. − 329. i)10-s + (331. − 416. i)11-s − 261.·12-s + (307. − 385. i)13-s + (273. − 1.19e3i)14-s + (195. − 854. i)15-s + (−692. + 868. i)16-s − 1.20e3·17-s + ⋯ |
| L(s) = 1 | + (0.307 + 1.34i)2-s + (0.520 + 0.250i)3-s + (−0.816 + 0.393i)4-s + (−0.387 − 1.69i)5-s + (−0.177 + 0.777i)6-s + (−1.09 − 0.525i)7-s + (0.0804 + 0.100i)8-s + (0.207 + 0.260i)9-s + (2.16 − 1.04i)10-s + (0.827 − 1.03i)11-s − 0.523·12-s + (0.504 − 0.632i)13-s + (0.372 − 1.63i)14-s + (0.223 − 0.980i)15-s + (−0.676 + 0.848i)16-s − 1.00·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.92157 - 0.196986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.92157 - 0.196986i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.10 - 3.90i)T \) |
| 29 | \( 1 + (1.86e3 - 4.12e3i)T \) |
| good | 2 | \( 1 + (-1.73 - 7.61i)T + (-28.8 + 13.8i)T^{2} \) |
| 5 | \( 1 + (21.6 + 94.9i)T + (-2.81e3 + 1.35e3i)T^{2} \) |
| 7 | \( 1 + (141. + 68.1i)T + (1.04e4 + 1.31e4i)T^{2} \) |
| 11 | \( 1 + (-331. + 416. i)T + (-3.58e4 - 1.57e5i)T^{2} \) |
| 13 | \( 1 + (-307. + 385. i)T + (-8.26e4 - 3.61e5i)T^{2} \) |
| 17 | \( 1 + 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-2.06e3 + 993. i)T + (1.54e6 - 1.93e6i)T^{2} \) |
| 23 | \( 1 + (-736. + 3.22e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 31 | \( 1 + (-4.20 - 18.4i)T + (-2.57e7 + 1.24e7i)T^{2} \) |
| 37 | \( 1 + (8.49e3 + 1.06e4i)T + (-1.54e7 + 6.76e7i)T^{2} \) |
| 41 | \( 1 - 1.74e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.40e3 + 6.15e3i)T + (-1.32e8 - 6.37e7i)T^{2} \) |
| 47 | \( 1 + (1.56e4 - 1.96e4i)T + (-5.10e7 - 2.23e8i)T^{2} \) |
| 53 | \( 1 + (-2.86e3 - 1.25e4i)T + (-3.76e8 + 1.81e8i)T^{2} \) |
| 59 | \( 1 - 2.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (1.86e4 + 8.98e3i)T + (5.26e8 + 6.60e8i)T^{2} \) |
| 67 | \( 1 + (403. + 506. i)T + (-3.00e8 + 1.31e9i)T^{2} \) |
| 71 | \( 1 + (7.62e3 - 9.55e3i)T + (-4.01e8 - 1.75e9i)T^{2} \) |
| 73 | \( 1 + (-9.92e3 + 4.34e4i)T + (-1.86e9 - 8.99e8i)T^{2} \) |
| 79 | \( 1 + (-1.11e4 - 1.39e4i)T + (-6.84e8 + 2.99e9i)T^{2} \) |
| 83 | \( 1 + (-5.11e4 + 2.46e4i)T + (2.45e9 - 3.07e9i)T^{2} \) |
| 89 | \( 1 + (-7.04e3 - 3.08e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 + (-1.56e5 + 7.52e4i)T + (5.35e9 - 6.71e9i)T^{2} \) |
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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
−13.38164718718737002714472995205, −12.69293638462836395078233303688, −11.01201690325796387019084865297, −9.130403790832831253253422558401, −8.715858819355142772205334668850, −7.44695248749170775085327637261, −6.13904389549240219628637175702, −4.87386154549808575154294483684, −3.68651706146448522357656502227, −0.68944349843357561520872324247,
1.92524760252129592153475265189, 3.10555006864053958593385117997, 3.87044536639504592555552389922, 6.50513695117059225139831391565, 7.30152593887674957313216075073, 9.400565837549969129654380273374, 10.01048465031955322344946757461, 11.39959924314715306734204415186, 11.89897197928585658685877993248, 13.17803236875061209865414829759
