L(s) = 1 | + (0.973 − 1.68i)2-s + (0.867 + 1.50i)3-s + (−0.895 − 1.55i)4-s + (−1.85 + 3.22i)5-s + 3.37·6-s + (1.95 − 1.78i)7-s + 0.406·8-s + (−0.00432 + 0.00749i)9-s + (3.62 + 6.27i)10-s + (1.78 + 3.08i)11-s + (1.55 − 2.69i)12-s + (−1.10 − 5.03i)14-s − 6.45·15-s + (2.18 − 3.78i)16-s + (−0.847 − 1.46i)17-s + (0.00843 + 0.0146i)18-s + ⋯ |
L(s) = 1 | + (0.688 − 1.19i)2-s + (0.500 + 0.867i)3-s + (−0.447 − 0.775i)4-s + (−0.831 + 1.44i)5-s + 1.37·6-s + (0.738 − 0.674i)7-s + 0.143·8-s + (−0.00144 + 0.00249i)9-s + (1.14 + 1.98i)10-s + (0.537 + 0.930i)11-s + (0.448 − 0.776i)12-s + (−0.295 − 1.34i)14-s − 1.66·15-s + (0.546 − 0.946i)16-s + (−0.205 − 0.355i)17-s + (0.00198 + 0.00344i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.780697228\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.780697228\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.95 + 1.78i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.973 + 1.68i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.867 - 1.50i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.85 - 3.22i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.78 - 3.08i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.847 + 1.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.74 - 4.75i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.09 - 5.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 + (-3.52 - 6.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.907 - 1.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.877T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + (1.36 - 2.36i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.02 + 3.50i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.60 - 2.78i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.24 + 12.5i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.16 + 7.21i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.11T + 71T^{2} \) |
| 73 | \( 1 + (-1.53 - 2.65i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.843 + 1.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.44T + 83T^{2} \) |
| 89 | \( 1 + (2.23 - 3.86i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−10.20274838925946746668329161273, −9.438855784096873687690417067692, −8.067389762135375850892680999629, −7.41164212206499852833141631302, −6.56786901204058401770019479924, −4.94692487787694544514304571402, −3.95710047014372978118853464949, −3.86803616754695010510756059645, −2.85312038701595620706571008493, −1.68518504020260420293447385790,
1.00185541996890218556423821084, 2.31207664443322178602535957811, 4.12739078356703461499400710040, 4.58824764723181763058950972796, 5.56877143446350260827992300448, 6.34634940292674931565574768173, 7.36282910504547165919171548275, 8.073070465992457983257540538569, 8.528399930793396648367603047275, 9.002791458288336408729098317840