L(s) = 1 | + (−0.950 + 1.04i)2-s + (1.36 − 2.37i)3-s + (−0.192 − 1.99i)4-s + (−0.866 + 0.5i)5-s + (1.18 + 3.68i)6-s + (−1.02 − 2.43i)7-s + (2.26 + 1.69i)8-s + (−2.24 − 3.89i)9-s + (0.299 − 1.38i)10-s + (0.0868 + 0.0501i)11-s + (−4.98 − 2.26i)12-s − 4.11i·13-s + (3.52 + 1.24i)14-s + 2.73i·15-s + (−3.92 + 0.766i)16-s + (4.67 + 2.69i)17-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.740i)2-s + (0.790 − 1.36i)3-s + (−0.0962 − 0.995i)4-s + (−0.387 + 0.223i)5-s + (0.482 + 1.50i)6-s + (−0.387 − 0.922i)7-s + (0.801 + 0.597i)8-s + (−0.748 − 1.29i)9-s + (0.0948 − 0.437i)10-s + (0.0261 + 0.0151i)11-s + (−1.43 − 0.654i)12-s − 1.14i·13-s + (0.942 + 0.333i)14-s + 0.706i·15-s + (−0.981 + 0.191i)16-s + (1.13 + 0.654i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.811902 - 0.417583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.811902 - 0.417583i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.950 - 1.04i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (1.02 + 2.43i)T \) |
good | 3 | \( 1 + (-1.36 + 2.37i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.0868 - 0.0501i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.11iT - 13T^{2} \) |
| 17 | \( 1 + (-4.67 - 2.69i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.72 - 6.45i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.30 + 0.754i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + (2.72 - 4.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.519 + 0.899i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.99iT - 41T^{2} \) |
| 43 | \( 1 + 7.04iT - 43T^{2} \) |
| 47 | \( 1 + (-2.22 - 3.84i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.07 - 5.31i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.26 + 7.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.84 + 3.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0950 + 0.0548i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.73iT - 71T^{2} \) |
| 73 | \( 1 + (5.32 + 3.07i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.70 + 2.13i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.50T + 83T^{2} \) |
| 89 | \( 1 + (2.76 - 1.59i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−13.16016231485467041353614730093, −12.29366445241778485217988935031, −10.68888175659962063833672196464, −9.762919244788860801803147489338, −8.247527785713931355364318451174, −7.74017069092279978508219660643, −6.96763827943764723412652806786, −5.74774595628150616684199815382, −3.39029276112199141427952699919, −1.23131842623866793757342007767,
2.68364643403115004492391290084, 3.76829500450089196327090485369, 5.04651717941627633690640738016, 7.31952043981650950930447790732, 8.687883475502797620204997102192, 9.279477480484963795300501472314, 9.868545632285551380015790362176, 11.25548506342399271575982456900, 11.91067687873637677864992685590, 13.23864699211808863816331132100