| L(s) = 1 | + (−0.893 + 0.448i)2-s + (−1.41 + 1.00i)3-s + (0.597 − 0.802i)4-s + (0.308 − 1.02i)5-s + (0.814 − 1.52i)6-s + (−2.06 + 4.78i)7-s + (−0.173 + 0.984i)8-s + (0.996 − 2.82i)9-s + (0.186 + 1.05i)10-s + (−5.51 − 1.30i)11-s + (−0.0413 + 1.73i)12-s + (−2.75 + 1.81i)13-s + (−0.303 − 5.20i)14-s + (0.594 + 1.76i)15-s + (−0.286 − 0.957i)16-s + (−1.32 − 0.483i)17-s + ⋯ |
| L(s) = 1 | + (−0.631 + 0.317i)2-s + (−0.816 + 0.577i)3-s + (0.298 − 0.401i)4-s + (0.137 − 0.460i)5-s + (0.332 − 0.624i)6-s + (−0.780 + 1.80i)7-s + (−0.0613 + 0.348i)8-s + (0.332 − 0.943i)9-s + (0.0589 + 0.334i)10-s + (−1.66 − 0.394i)11-s + (−0.0119 + 0.499i)12-s + (−0.765 + 0.503i)13-s + (−0.0810 − 1.39i)14-s + (0.153 + 0.455i)15-s + (−0.0717 − 0.239i)16-s + (−0.322 − 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0263459 + 0.303633i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0263459 + 0.303633i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.893 - 0.448i)T \) |
| 3 | \( 1 + (1.41 - 1.00i)T \) |
| good | 5 | \( 1 + (-0.308 + 1.02i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (2.06 - 4.78i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (5.51 + 1.30i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (2.75 - 1.81i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (1.32 + 0.483i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (0.526 - 0.191i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.185 - 0.429i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (0.413 - 7.09i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-2.69 - 0.314i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (3.28 - 2.75i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-5.78 - 2.90i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (-6.69 + 7.09i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-1.37 + 0.161i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (3.96 + 6.86i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.06 - 0.252i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (-5.03 - 6.75i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (0.100 + 1.73i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (-1.32 - 7.51i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.86 - 10.5i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.99 - 1.00i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (10.3 - 5.19i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (0.973 - 5.52i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (0.471 + 1.57i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−12.93061333751946483724987958034, −12.31394383908970843953410656110, −11.23607357362685833259232211919, −10.15593662633029151277558754940, −9.268204344142566662992600545311, −8.540291064299429644134702000723, −6.88378759398285987122647463733, −5.64414699627172635435127476983, −5.10913381130699535969531624078, −2.70153864666891934332298071249,
0.36359044079345667058916013083, 2.65001148016529527745814628992, 4.55214756944611784107301805090, 6.24009083529781515923218063099, 7.34988545036762486303079494445, 7.76426259301668671046459346924, 9.873064459931335105941212770475, 10.47014406506065171973420868364, 10.99323342581375986068066223051, 12.54591764315023488682955971645
