| L(s) = 1 | + (10.1 − 2.71i)2-s + (−5.75 + 21.4i)3-s + (67.6 − 39.0i)4-s + (−25.1 + 49.9i)5-s + 233. i·6-s + (112. − 63.7i)7-s + (342. − 342. i)8-s + (−217. − 125. i)9-s + (−119. + 574. i)10-s + (−5.52 − 9.57i)11-s + (449. + 1.67e3i)12-s + (−500. − 500. i)13-s + (971. − 952. i)14-s + (−926. − 827. i)15-s + (1.28e3 − 2.23e3i)16-s + (−851. − 228. i)17-s + ⋯ |
| L(s) = 1 | + (1.79 − 0.480i)2-s + (−0.368 + 1.37i)3-s + (2.11 − 1.22i)4-s + (−0.450 + 0.892i)5-s + 2.64i·6-s + (0.870 − 0.491i)7-s + (1.88 − 1.88i)8-s + (−0.894 − 0.516i)9-s + (−0.377 + 1.81i)10-s + (−0.0137 − 0.0238i)11-s + (0.900 + 3.36i)12-s + (−0.820 − 0.820i)13-s + (1.32 − 1.29i)14-s + (−1.06 − 0.949i)15-s + (1.25 − 2.17i)16-s + (−0.714 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.51266 + 0.848391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.51266 + 0.848391i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (25.1 - 49.9i)T \) |
| 7 | \( 1 + (-112. + 63.7i)T \) |
| good | 2 | \( 1 + (-10.1 + 2.71i)T + (27.7 - 16i)T^{2} \) |
| 3 | \( 1 + (5.75 - 21.4i)T + (-210. - 121.5i)T^{2} \) |
| 11 | \( 1 + (5.52 + 9.57i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (500. + 500. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (851. + 228. i)T + (1.22e6 + 7.09e5i)T^{2} \) |
| 19 | \( 1 + (-1.11e3 + 1.92e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-469. - 1.75e3i)T + (-5.57e6 + 3.21e6i)T^{2} \) |
| 29 | \( 1 - 3.82e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.31e3 + 757. i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-3.48e3 + 934. i)T + (6.00e7 - 3.46e7i)T^{2} \) |
| 41 | \( 1 - 1.30e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (1.33e4 - 1.33e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (672. + 2.50e3i)T + (-1.98e8 + 1.14e8i)T^{2} \) |
| 53 | \( 1 + (-1.26e4 - 3.37e3i)T + (3.62e8 + 2.09e8i)T^{2} \) |
| 59 | \( 1 + (5.38e3 + 9.32e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-3.53e4 - 2.04e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.98e3 + 7.40e3i)T + (-1.16e9 - 6.75e8i)T^{2} \) |
| 71 | \( 1 + 7.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.14e4 - 4.26e4i)T + (-1.79e9 - 1.03e9i)T^{2} \) |
| 79 | \( 1 + (6.46e4 + 3.73e4i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.35e4 - 3.35e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + (7.06e3 - 1.22e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (1.49e4 - 1.49e4i)T - 8.58e9iT^{2} \) |
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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
−15.16416808506961251123021120958, −14.65433334204795218299284943384, −13.38296271883013480727709345559, −11.61599780971054792918443063119, −11.06484142729429149986398614282, −10.11002535080861321148822690264, −7.15523056997078052633597945554, −5.27325803029009050608491314656, −4.36234519596454941752915835033, −3.00723579087168901869699141648,
1.94749304615409918808939308709, 4.47999567023943261809976556534, 5.72178224796673139076529318563, 7.10039368057134577031141786341, 8.183565019086321479482240013722, 11.69326320426179037767465417438, 12.02012191032714738468135566800, 12.98935079965271982556328404263, 14.00917909985788115789204323942, 15.10120388527302434207625074922
