| L(s) = 1 | + (4.55 + 7.89i)7-s + (0.383 − 0.221i)11-s + (−5.55 + 9.62i)13-s − 8.01i·17-s − 8.11·19-s + (20.4 + 11.8i)23-s + (45.9 − 26.5i)29-s + (−14.6 + 25.4i)31-s − 18.4·37-s + (38.9 + 22.4i)41-s + (11.5 + 19.9i)43-s + (−7.32 + 4.22i)47-s + (−17.0 + 29.5i)49-s + 60.5i·53-s + (−65.9 − 38.0i)59-s + ⋯ |
| L(s) = 1 | + (0.651 + 1.12i)7-s + (0.0348 − 0.0201i)11-s + (−0.427 + 0.740i)13-s − 0.471i·17-s − 0.427·19-s + (0.888 + 0.513i)23-s + (1.58 − 0.913i)29-s + (−0.473 + 0.819i)31-s − 0.499·37-s + (0.950 + 0.548i)41-s + (0.267 + 0.463i)43-s + (−0.155 + 0.0899i)47-s + (−0.348 + 0.602i)49-s + 1.14i·53-s + (−1.11 − 0.645i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.728676718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.728676718\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-4.55 - 7.89i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.383 + 0.221i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (5.55 - 9.62i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 8.01iT - 289T^{2} \) |
| 19 | \( 1 + 8.11T + 361T^{2} \) |
| 23 | \( 1 + (-20.4 - 11.8i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-45.9 + 26.5i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (14.6 - 25.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 18.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-38.9 - 22.4i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-11.5 - 19.9i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (7.32 - 4.22i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 60.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (2.67 + 4.63i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (54.8 - 95.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 4.35T + 5.32e3T^{2} \) |
| 79 | \( 1 + (0.792 + 1.37i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (7.32 - 4.22i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-57.6 - 99.7i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−8.971465692524241472932867165721, −8.230807681115473017716851074441, −7.43362605422936587813748106227, −6.59012909905085534093315550003, −5.80638321569557161697490527046, −4.94189307485642114490121401196, −4.39222669205570675509866628936, −3.03439709820427567177225794398, −2.30384655367070983033619381029, −1.24517119129622690467979791709,
0.41960360744156773356366048875, 1.41498756638500284366834422796, 2.62534662200596472271823333164, 3.65053582224456278113652295619, 4.51255161516770386374541580194, 5.13537334992605735848967407063, 6.17983040354082722382017603030, 7.02666501365867270666623903373, 7.62977796460519564838540868456, 8.345579187307383509313583241553
