| L(s) = 1 | + (1.62 + 1.36i)2-s + (−1.45 − 0.837i)3-s + (0.435 + 2.47i)4-s + (2.43 + 0.212i)5-s + (−1.21 − 3.34i)6-s + (−4.73 + 1.26i)7-s + (−0.541 + 0.937i)8-s + (−0.0964 − 0.167i)9-s + (3.66 + 3.66i)10-s + (0.312 − 3.57i)11-s + (1.43 − 3.95i)12-s + (0.0362 − 0.0777i)13-s + (−9.42 − 4.39i)14-s + (−3.35 − 2.34i)15-s + (2.55 − 0.930i)16-s + (−1.56 + 5.85i)17-s + ⋯ |
| L(s) = 1 | + (1.15 + 0.965i)2-s + (−0.837 − 0.483i)3-s + (0.217 + 1.23i)4-s + (1.08 + 0.0952i)5-s + (−0.496 − 1.36i)6-s + (−1.78 + 0.479i)7-s + (−0.191 + 0.331i)8-s + (−0.0321 − 0.0556i)9-s + (1.15 + 1.15i)10-s + (0.0942 − 1.07i)11-s + (0.415 − 1.14i)12-s + (0.0100 − 0.0215i)13-s + (−2.51 − 1.17i)14-s + (−0.865 − 0.606i)15-s + (0.639 − 0.232i)16-s + (−0.380 + 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15808 + 0.514216i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15808 + 0.514216i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 + (4.00 + 7.54i)T \) |
| good | 2 | \( 1 + (-1.62 - 1.36i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (1.45 + 0.837i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.43 - 0.212i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (4.73 - 1.26i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.312 + 3.57i)T + (-10.8 - 1.91i)T^{2} \) |
| 13 | \( 1 + (-0.0362 + 0.0777i)T + (-8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (1.56 - 5.85i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.294 + 0.351i)T + (-3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-2.10 - 2.51i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.26 - 0.197i)T + (28.5 - 5.03i)T^{2} \) |
| 31 | \( 1 + (-1.57 - 2.24i)T + (-10.6 + 29.1i)T^{2} \) |
| 37 | \( 1 + (-5.69 + 4.77i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (9.71 + 3.53i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.554 - 2.06i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-4.53 + 2.11i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.279 + 3.18i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-4.78 - 2.23i)T + (37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (2.80 - 7.69i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.136 + 0.374i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.02 + 1.69i)T + (12.3 + 69.9i)T^{2} \) |
| 79 | \( 1 + (4.39 + 12.0i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (10.7 + 10.7i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.20 - 0.439i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.41 + 3.12i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.71472324013656126637931701505, −13.31183171700920137870476313943, −13.13335517319425906529510700552, −12.00605781005494011905301446606, −10.35584241849231714093002399323, −9.007467995253355750139443954502, −6.87529233833983887864469954982, −5.99176044889505243502204033627, −5.77867574508706598000807423247, −3.41011732265345443019754681243,
2.69741403199081772661178350195, 4.43349511542813697695589127079, 5.58748270625371103469889302710, 6.73580905966274904671741674741, 9.687418379164108309821421677301, 10.06525250851043860627425913298, 11.30117432959910547805342939385, 12.43362295943243892034043984064, 13.28308273120528012643941469099, 13.91030003947692621099174935393
