| L(s) = 1 | + (−1.69 − 3.32i)2-s + (−0.0858 − 0.541i)3-s + (−5.84 + 8.04i)4-s + (2.26 − 4.45i)5-s + (−1.65 + 1.20i)6-s + (1.68 − 1.68i)7-s + (21.9 + 3.47i)8-s + (8.27 − 2.68i)9-s + (−18.6 + 0.00137i)10-s + (−2.56 + 7.89i)11-s + (4.86 + 2.47i)12-s + (−5.04 + 9.90i)13-s + (−8.46 − 2.74i)14-s + (−2.60 − 0.847i)15-s + (−13.3 − 40.9i)16-s + (0.715 − 4.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.847 − 1.66i)2-s + (−0.0286 − 0.180i)3-s + (−1.46 + 2.01i)4-s + (0.453 − 0.891i)5-s + (−0.276 + 0.200i)6-s + (0.240 − 0.240i)7-s + (2.73 + 0.433i)8-s + (0.919 − 0.298i)9-s + (−1.86 + 0.000137i)10-s + (−0.233 + 0.717i)11-s + (0.405 + 0.206i)12-s + (−0.388 + 0.762i)13-s + (−0.604 − 0.196i)14-s + (−0.173 − 0.0565i)15-s + (−0.832 − 2.56i)16-s + (0.0421 − 0.265i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.276004 - 0.586427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.276004 - 0.586427i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.26 + 4.45i)T \) |
| good | 2 | \( 1 + (1.69 + 3.32i)T + (-2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (0.0858 + 0.541i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-1.68 + 1.68i)T - 49iT^{2} \) |
| 11 | \( 1 + (2.56 - 7.89i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (5.04 - 9.90i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-0.715 + 4.51i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-12.8 - 17.6i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (20.0 - 10.2i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-17.4 + 24.0i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (17.6 - 12.7i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-9.49 - 4.83i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (7.33 + 22.5i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-14.6 - 14.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (55.8 - 8.85i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-4.98 - 31.4i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (9.66 - 3.13i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-11.3 + 34.8i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (13.3 - 84.3i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (56.9 + 41.3i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-34.3 + 17.4i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-75.3 + 103. i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-68.5 - 10.8i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (152. + 49.4i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (46.9 - 7.44i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−17.50979429680325468593133685309, −16.26086719643544545913867635326, −13.80299787371794598752389725592, −12.60596677088728929252859558072, −11.87017252158552841161428343979, −10.10736454861473143184703646823, −9.400981584874086881904230583214, −7.78044642168231311241603787305, −4.36044784137580749560376181364, −1.65130694169913481180773155259,
5.35329919064720532055072637986, 6.80531082381916217231866069692, 8.041793492166365595705253283583, 9.646956331068100032976668700071, 10.67086028977398208948548576328, 13.43064158727244101207457365841, 14.60554546526631609539845192474, 15.52708096133780642714788374335, 16.50363103171067950271380125141, 17.91167151982323979937003154312
