| L(s) = 1 | + (−1.08 + 1.41i)2-s + (0.543 − 0.0356i)3-s + (−0.309 − 1.15i)4-s + (2.53 − 0.861i)5-s + (−0.541 + 0.810i)6-s + (−1.32 − 0.549i)8-s + (−2.67 + 0.352i)9-s + (−1.54 + 4.53i)10-s + (−2.21 − 4.49i)11-s + (−0.209 − 0.618i)12-s + (4.70 − 4.70i)13-s + (1.35 − 0.559i)15-s + (4.29 − 2.48i)16-s + (−4.12 + 0.0371i)17-s + (2.41 − 4.18i)18-s + (−4.21 − 3.23i)19-s + ⋯ |
| L(s) = 1 | + (−0.769 + 1.00i)2-s + (0.314 − 0.0205i)3-s + (−0.154 − 0.578i)4-s + (1.13 − 0.385i)5-s + (−0.221 + 0.330i)6-s + (−0.468 − 0.194i)8-s + (−0.893 + 0.117i)9-s + (−0.487 + 1.43i)10-s + (−0.668 − 1.35i)11-s + (−0.0605 − 0.178i)12-s + (1.30 − 1.30i)13-s + (0.348 − 0.144i)15-s + (1.07 − 0.620i)16-s + (−0.999 + 0.00900i)17-s + (0.569 − 0.986i)18-s + (−0.968 − 0.742i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.861837 - 0.300249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.861837 - 0.300249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 + (4.12 - 0.0371i)T \) |
| good | 2 | \( 1 + (1.08 - 1.41i)T + (-0.517 - 1.93i)T^{2} \) |
| 3 | \( 1 + (-0.543 + 0.0356i)T + (2.97 - 0.391i)T^{2} \) |
| 5 | \( 1 + (-2.53 + 0.861i)T + (3.96 - 3.04i)T^{2} \) |
| 11 | \( 1 + (2.21 + 4.49i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-4.70 + 4.70i)T - 13iT^{2} \) |
| 19 | \( 1 + (4.21 + 3.23i)T + (4.91 + 18.3i)T^{2} \) |
| 23 | \( 1 + (0.267 - 4.08i)T + (-22.8 - 3.00i)T^{2} \) |
| 29 | \( 1 + (-1.90 + 9.55i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (-0.0502 - 0.766i)T + (-30.7 + 4.04i)T^{2} \) |
| 37 | \( 1 + (5.32 + 2.62i)T + (22.5 + 29.3i)T^{2} \) |
| 41 | \( 1 + (0.0510 + 0.256i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.397 + 0.959i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (8.29 + 2.22i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.81 - 0.238i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-3.63 - 4.73i)T + (-15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-0.999 - 1.13i)T + (-7.96 + 60.4i)T^{2} \) |
| 67 | \( 1 + (-10.7 - 6.18i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.95 - 2.64i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.852 - 0.747i)T + (9.52 + 72.3i)T^{2} \) |
| 79 | \( 1 + (-12.0 - 0.791i)T + (78.3 + 10.3i)T^{2} \) |
| 83 | \( 1 + (-5.11 - 12.3i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.507 + 1.89i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-9.84 - 1.95i)T + (89.6 + 37.1i)T^{2} \) |
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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
−9.837100890223482827617213479478, −8.826824366198167431620624129798, −8.526572045082558233394337062871, −7.917320514379829001529608863281, −6.52770883322105683459383224453, −5.85572943497062337628069030167, −5.40911716302540493890074926971, −3.49544075485511270666516144146, −2.43953636177891056401440588402, −0.51544826059873441611238938668,
1.84919837282171490599910430846, 2.20880771927885466984607409973, 3.47099457393193835301888024347, 4.87323744249429455287239336445, 6.18891301144306357951836594873, 6.65881132546786261200926095825, 8.269890771662612394356061375054, 8.881516665877886889252605361564, 9.494525806346691742780929173231, 10.40445260930201648251003925423
