| L(s) = 1 | + (−0.900 − 0.433i)2-s + (−0.124 + 1.66i)3-s + (0.623 + 0.781i)4-s + (−0.955 − 0.294i)5-s + (0.834 − 1.44i)6-s + (−1.34 − 2.32i)7-s + (−0.222 − 0.974i)8-s + (0.215 + 0.0324i)9-s + (0.733 + 0.680i)10-s + (2.10 − 2.63i)11-s + (−1.37 + 0.939i)12-s + (1.60 − 1.48i)13-s + (0.200 + 2.67i)14-s + (0.609 − 1.55i)15-s + (−0.222 + 0.974i)16-s + (1.17 − 0.363i)17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.306i)2-s + (−0.0719 + 0.960i)3-s + (0.311 + 0.390i)4-s + (−0.427 − 0.131i)5-s + (0.340 − 0.589i)6-s + (−0.507 − 0.879i)7-s + (−0.0786 − 0.344i)8-s + (0.0717 + 0.0108i)9-s + (0.231 + 0.215i)10-s + (0.633 − 0.794i)11-s + (−0.397 + 0.271i)12-s + (0.444 − 0.412i)13-s + (0.0536 + 0.715i)14-s + (0.157 − 0.400i)15-s + (−0.0556 + 0.243i)16-s + (0.285 − 0.0880i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.950134 - 0.123712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.950134 - 0.123712i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.955 + 0.294i)T \) |
| 43 | \( 1 + (-5.44 + 3.65i)T \) |
| good | 3 | \( 1 + (0.124 - 1.66i)T + (-2.96 - 0.447i)T^{2} \) |
| 7 | \( 1 + (1.34 + 2.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.10 + 2.63i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.60 + 1.48i)T + (0.971 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.17 + 0.363i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-3.67 + 0.553i)T + (18.1 - 5.60i)T^{2} \) |
| 23 | \( 1 + (-0.622 - 1.58i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.669 - 8.93i)T + (-28.6 + 4.32i)T^{2} \) |
| 31 | \( 1 + (-6.32 + 4.31i)T + (11.3 - 28.8i)T^{2} \) |
| 37 | \( 1 + (-1.55 + 2.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.414 - 0.199i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (5.66 + 7.10i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (5.47 + 5.07i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-2.08 + 9.13i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-5.77 - 3.93i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (-2.99 + 0.451i)T + (64.0 - 19.7i)T^{2} \) |
| 71 | \( 1 + (-2.21 + 5.63i)T + (-52.0 - 48.2i)T^{2} \) |
| 73 | \( 1 + (3.84 - 3.57i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (2.99 + 5.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.503 - 6.72i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (0.220 - 2.93i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (2.10 - 2.64i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.95229527543603993235148956131, −10.18690549430427821455085255953, −9.496836857430154134764682028541, −8.614481995174253737174452930710, −7.56169805082469039581094830834, −6.61935930070751088153108071560, −5.20094793258098175055615947000, −3.87193114256676161748203768032, −3.33042675814682900037211033555, −0.948616167709183787038569402707,
1.28066283610278780939296702674, 2.69264900722469589220708438373, 4.38754062097920959030187151588, 6.00021129562601058413184378306, 6.61450449342441777342802468854, 7.50424158200531259057605233674, 8.306576787124111373812890918296, 9.374824705324951925484071406430, 10.01902078375272670039789673483, 11.42948756428825888767073528524
