L(s) = 1 | + (0.349 − 1.37i)2-s + (−1.72 − 0.203i)3-s + (−1.75 − 0.958i)4-s − 2.27i·5-s + (−0.880 + 2.28i)6-s − i·7-s + (−1.92 + 2.06i)8-s + (2.91 + 0.699i)9-s + (−3.12 − 0.796i)10-s + 4.31·11-s + (2.82 + 2.00i)12-s − 0.406·13-s + (−1.37 − 0.349i)14-s + (−0.463 + 3.91i)15-s + (2.16 + 3.36i)16-s + 4.31i·17-s + ⋯ |
L(s) = 1 | + (0.247 − 0.968i)2-s + (−0.993 − 0.117i)3-s + (−0.877 − 0.479i)4-s − 1.01i·5-s + (−0.359 + 0.933i)6-s − 0.377i·7-s + (−0.681 + 0.731i)8-s + (0.972 + 0.233i)9-s + (−0.986 − 0.251i)10-s + 1.30·11-s + (0.815 + 0.579i)12-s − 0.112·13-s + (−0.366 − 0.0934i)14-s + (−0.119 + 1.01i)15-s + (0.540 + 0.841i)16-s + 1.04i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.355215 - 0.687979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.355215 - 0.687979i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.349 + 1.37i)T \) |
| 3 | \( 1 + (1.72 + 0.203i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.27iT - 5T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 + 0.406T + 13T^{2} \) |
| 17 | \( 1 - 4.31iT - 17T^{2} \) |
| 19 | \( 1 + 5.42iT - 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 + 3.72iT - 29T^{2} \) |
| 31 | \( 1 - 5.83iT - 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 - 2.56iT - 41T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 - 2.32T + 47T^{2} \) |
| 53 | \( 1 + 5.48iT - 53T^{2} \) |
| 59 | \( 1 - 3.91T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 7.83iT - 67T^{2} \) |
| 71 | \( 1 + 4.88T + 71T^{2} \) |
| 73 | \( 1 - 2.81T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 0.641T + 83T^{2} \) |
| 89 | \( 1 - 13.9iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−13.42698639863552038067355066484, −12.65252900996271355841378805107, −11.81320311044038914048390802957, −10.90189492122606933178466181900, −9.707625398429553857920881174750, −8.610006673338277973169977923694, −6.59647803541515862415146053023, −5.13072754573617094397881489375, −4.10845721088601200187235172368, −1.23268990940303798440357755413,
3.79170952090372095817767301368, 5.39540967448921631631149239044, 6.51656590076277413526604915996, 7.28633647865419757907446770569, 9.069501378673835017535979028270, 10.24316535123926160564823581734, 11.62531553815766633727238088397, 12.40727709113359965946896500485, 13.92542886168324957774443654594, 14.72991515149737289663560296749