| L(s) = 1 | + (−2.08 + 1.51i)2-s + (0.913 − 0.406i)3-s + (1.43 − 4.42i)4-s + (−1.93 − 3.34i)5-s + (−1.29 + 2.23i)6-s + (−2.07 − 2.30i)7-s + (2.11 + 6.52i)8-s + (0.669 − 0.743i)9-s + (9.10 + 4.05i)10-s + (1.06 − 0.226i)11-s + (−0.486 − 4.63i)12-s + (−0.249 + 2.36i)13-s + (7.82 + 1.66i)14-s + (−3.12 − 2.27i)15-s + (−6.77 − 4.92i)16-s + (1.60 + 0.342i)17-s + ⋯ |
| L(s) = 1 | + (−1.47 + 1.07i)2-s + (0.527 − 0.234i)3-s + (0.719 − 2.21i)4-s + (−0.864 − 1.49i)5-s + (−0.526 + 0.912i)6-s + (−0.784 − 0.871i)7-s + (0.749 + 2.30i)8-s + (0.223 − 0.247i)9-s + (2.88 + 1.28i)10-s + (0.320 − 0.0681i)11-s + (−0.140 − 1.33i)12-s + (−0.0690 + 0.657i)13-s + (2.09 + 0.444i)14-s + (−0.807 − 0.586i)15-s + (−1.69 − 1.23i)16-s + (0.390 + 0.0829i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.414984 - 0.185294i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.414984 - 0.185294i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 31 | \( 1 + (-3.03 + 4.66i)T \) |
| good | 2 | \( 1 + (2.08 - 1.51i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (1.93 + 3.34i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.07 + 2.30i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 0.226i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.249 - 2.36i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-1.60 - 0.342i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.0767 + 0.730i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.222 + 0.685i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.88 + 5.73i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-1.16 + 2.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.77 - 1.68i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.791 - 7.53i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (1.30 + 0.947i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.07 - 2.30i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (9.44 - 4.20i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + (1.32 + 2.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.02 - 5.58i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-1.71 + 0.364i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (10.7 + 2.29i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-9.96 - 4.43i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-1.89 + 5.84i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.74 + 11.5i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−14.15818915859899966613982457443, −12.97374551651304419454094848896, −11.65170672582563563677368205453, −10.00753981914160417102491844989, −9.201330736669419202999692584585, −8.281214597041239951115484486361, −7.50074073644890746814588432827, −6.30669253897930161097719166833, −4.36333469471639422167423279654, −0.864884957248952748652388981828,
2.73706221354807714501593820411, 3.42400127481338135360562000459, 6.74256899135794711688566228226, 7.85105647895783705177178523795, 8.851244718068371834083542664464, 10.01908064090618127567288386565, 10.65201665198404925290902974765, 11.79145690617326077678362678769, 12.53796536106329055356311388402, 14.31890412379382491758606805262
