L(s) = 1 | + (−0.683 − 2.55i)2-s + (0.839 − 1.51i)3-s + (−4.30 + 2.48i)4-s + (−0.283 + 1.05i)5-s + (−4.43 − 1.10i)6-s + (−1.64 − 2.85i)7-s + (5.55 + 5.55i)8-s + (−1.58 − 2.54i)9-s + 2.89·10-s + 1.47·11-s + (0.150 + 8.61i)12-s + (3.81 + 1.02i)13-s + (−6.16 + 6.16i)14-s + (1.36 + 1.31i)15-s + (5.39 − 9.33i)16-s + (6.35 − 1.70i)17-s + ⋯ |
L(s) = 1 | + (−0.483 − 1.80i)2-s + (0.484 − 0.874i)3-s + (−2.15 + 1.24i)4-s + (−0.126 + 0.473i)5-s + (−1.81 − 0.451i)6-s + (−0.623 − 1.07i)7-s + (1.96 + 1.96i)8-s + (−0.529 − 0.848i)9-s + 0.915·10-s + 0.444·11-s + (0.0434 + 2.48i)12-s + (1.05 + 0.283i)13-s + (−1.64 + 1.64i)14-s + (0.352 + 0.340i)15-s + (1.34 − 2.33i)16-s + (1.54 − 0.413i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0268158 + 0.792910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0268158 + 0.792910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.839 + 1.51i)T \) |
| 37 | \( 1 + (2.43 - 5.57i)T \) |
good | 2 | \( 1 + (0.683 + 2.55i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (0.283 - 1.05i)T + (-4.33 - 2.5i)T^{2} \) |
| 7 | \( 1 + (1.64 + 2.85i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + (-3.81 - 1.02i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-6.35 + 1.70i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.74 + 1.27i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (2.17 + 2.17i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.685 - 0.685i)T - 29iT^{2} \) |
| 31 | \( 1 + (-1.98 - 1.98i)T + 31iT^{2} \) |
| 41 | \( 1 + (-2.55 - 4.42i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.78 + 2.78i)T - 43iT^{2} \) |
| 47 | \( 1 - 6.97iT - 47T^{2} \) |
| 53 | \( 1 + (1.68 + 0.975i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.35 + 1.43i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (0.141 - 0.526i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (6.22 - 3.59i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.74 - 1.58i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 0.709iT - 73T^{2} \) |
| 79 | \( 1 + (-7.13 - 1.91i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-12.7 - 7.37i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.26 + 12.1i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.53 + 2.53i)T - 97iT^{2} \) |
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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
−12.93078976261645617707606686842, −12.04670386961178793090251829734, −11.02083819076341682405984868619, −10.13688361709161379487088450115, −9.048269098112056155182228078568, −7.972102288130768192834327159543, −6.65364859300555166858401928683, −3.93017628078033332703435243264, −3.00146600663723215958012812735, −1.16174943886522158047001995262,
3.90115498925025437761126477261, 5.43684271475029158307385824838, 6.17643097476195863484919833269, 7.938338151255012567652326069741, 8.690373795457559610514157572764, 9.368077325480789208974749239747, 10.43818887243817782553189455752, 12.41653980138760603828264726853, 13.65302168223783440810239039163, 14.68984504226460510533024943902