| L(s) = 1 | + (2.64 + 1.27i)2-s + (−4.95 + 6.21i)3-s + (0.375 + 0.470i)4-s + (13.7 + 6.63i)5-s + (−20.9 + 10.1i)6-s + (6.54 − 8.20i)7-s + (−4.82 − 21.1i)8-s + (−8.03 − 35.2i)9-s + (27.9 + 35.0i)10-s + (7.69 − 33.7i)11-s − 4.78·12-s + (−13.6 + 59.8i)13-s + (27.7 − 13.3i)14-s + (−109. + 52.7i)15-s + (15.2 − 66.7i)16-s + 62.6·17-s + ⋯ |
| L(s) = 1 | + (0.934 + 0.449i)2-s + (−0.953 + 1.19i)3-s + (0.0469 + 0.0588i)4-s + (1.23 + 0.593i)5-s + (−1.42 + 0.687i)6-s + (0.353 − 0.443i)7-s + (−0.213 − 0.934i)8-s + (−0.297 − 1.30i)9-s + (0.884 + 1.10i)10-s + (0.210 − 0.923i)11-s − 0.115·12-s + (−0.291 + 1.27i)13-s + (0.529 − 0.254i)14-s + (−1.88 + 0.907i)15-s + (0.238 − 1.04i)16-s + 0.893·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.23591 + 0.943275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23591 + 0.943275i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-113. + 107. i)T \) |
| good | 2 | \( 1 + (-2.64 - 1.27i)T + (4.98 + 6.25i)T^{2} \) |
| 3 | \( 1 + (4.95 - 6.21i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-13.7 - 6.63i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (-6.54 + 8.20i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (-7.69 + 33.7i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (13.6 - 59.8i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 - 62.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (48.8 + 61.2i)T + (-1.52e3 + 6.68e3i)T^{2} \) |
| 23 | \( 1 + (186. - 89.7i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (-36.3 - 17.5i)T + (1.85e4 + 2.32e4i)T^{2} \) |
| 37 | \( 1 + (7.36 + 32.2i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-316. + 152. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (23.8 - 104. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-56.6 - 27.2i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + 28.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-74.7 + 93.6i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-64.8 - 284. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (194. - 851. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-412. + 198. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + (-15.4 - 67.5i)T + (-4.44e5 + 2.13e5i)T^{2} \) |
| 83 | \( 1 + (-408. - 512. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (422. + 203. i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (-208. - 261. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
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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
−16.69171336616307578484616994787, −15.63968107923620537006967594841, −14.24345213229055755362912367385, −13.79053576144318524779858440994, −11.75131968980933270679989070455, −10.40465496435047204664926504997, −9.558767479143486629782466377613, −6.45697239200975191726444098621, −5.53321302806581299263192021325, −4.13552979945482495660795443139,
1.93765168427981809615247289950, 5.13483236098682443401884983380, 6.05335607082790633502339752311, 8.128597164509377533548545554540, 10.24054493154071610412772433239, 12.20758194756253963127787871240, 12.38452715658235136879005349207, 13.47057982518323073300897144537, 14.62963850527024994806056390731, 16.86893413809948800269104440783
