L(s) = 1 | + (−1.08 + 0.903i)2-s + (0.366 − 1.96i)4-s − 1.59·5-s + 1.43i·7-s + (1.37 + 2.46i)8-s + (1.73 − 1.43i)10-s + 4.93i·11-s + 5.37i·13-s + (−1.30 − 1.56i)14-s + (−3.73 − 1.43i)16-s − 1.32i·17-s + 0.267·19-s + (−0.582 + 3.13i)20-s + (−4.46 − 5.37i)22-s + 5.94·23-s + ⋯ |
L(s) = 1 | + (−0.769 + 0.639i)2-s + (0.183 − 0.983i)4-s − 0.712·5-s + 0.544i·7-s + (0.487 + 0.873i)8-s + (0.547 − 0.455i)10-s + 1.48i·11-s + 1.48i·13-s + (−0.347 − 0.418i)14-s + (−0.933 − 0.359i)16-s − 0.320i·17-s + 0.0614·19-s + (−0.130 + 0.700i)20-s + (−0.951 − 1.14i)22-s + 1.23·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.319586 + 0.544525i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.319586 + 0.544525i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.08 - 0.903i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 - 1.43iT - 7T^{2} \) |
| 11 | \( 1 - 4.93iT - 11T^{2} \) |
| 13 | \( 1 - 5.37iT - 13T^{2} \) |
| 17 | \( 1 + 1.32iT - 17T^{2} \) |
| 19 | \( 1 - 0.267T + 19T^{2} \) |
| 23 | \( 1 - 5.94T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 - 7.86iT - 31T^{2} \) |
| 37 | \( 1 + 2.49iT - 37T^{2} \) |
| 41 | \( 1 + 9.87iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 9.12T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 - 4.93iT - 59T^{2} \) |
| 61 | \( 1 + 5.37iT - 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 + 7.92T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 7.23iT - 83T^{2} \) |
| 89 | \( 1 - 15.7iT - 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−12.37860880059795683390758829882, −11.61896592108409415733949520740, −10.59476071122114440949650835507, −9.338551552616144430247887099972, −8.885030905993442203190376787472, −7.38026280028054756113746019690, −7.00866961954240113315989820163, −5.45576988794304963841799282607, −4.27645062418894776376554348513, −2.01283273142159126368758006683,
0.69815607724293954075145359515, 3.02127158287855238749978511372, 3.96840395169755656492002567578, 5.80685268189574143665793207670, 7.41533115507032009765001414462, 8.045113798665173744380761630439, 9.009423590532349559331410401821, 10.24036118506365217164833525112, 11.05404685353556287503376192648, 11.62278308529357930621827206651