L(s) = 1 | + (1.02 + 0.977i)2-s + (−1.35 − 1.07i)3-s + (0.0884 + 1.99i)4-s + (−2.18 − 1.26i)5-s + (−0.335 − 2.42i)6-s + (1.10 − 0.637i)7-s + (−1.86 + 2.12i)8-s + (0.686 + 2.92i)9-s + (−1 − 3.42i)10-s + (0.252 + 0.437i)11-s + (2.02 − 2.80i)12-s + (1.18 − 2.05i)13-s + (1.75 + 0.428i)14-s + (1.61 + 4.06i)15-s + (−3.98 + 0.353i)16-s + 0.792i·17-s + ⋯ |
L(s) = 1 | + (0.722 + 0.691i)2-s + (−0.783 − 0.621i)3-s + (0.0442 + 0.999i)4-s + (−0.977 − 0.564i)5-s + (−0.137 − 0.990i)6-s + (0.417 − 0.241i)7-s + (−0.658 + 0.752i)8-s + (0.228 + 0.973i)9-s + (−0.316 − 1.08i)10-s + (0.0761 + 0.131i)11-s + (0.585 − 0.810i)12-s + (0.328 − 0.569i)13-s + (0.468 + 0.114i)14-s + (0.415 + 1.04i)15-s + (−0.996 + 0.0883i)16-s + 0.192i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.767327 + 0.175428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.767327 + 0.175428i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.02 - 0.977i)T \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
good | 5 | \( 1 + (2.18 + 1.26i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.10 + 0.637i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.252 - 0.437i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.18 + 2.05i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.792iT - 17T^{2} \) |
| 19 | \( 1 - 4.70iT - 19T^{2} \) |
| 23 | \( 1 + (-1.61 + 2.78i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.18 + 1.26i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.04 + 4.06i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.74T + 37T^{2} \) |
| 41 | \( 1 + (-5.87 - 3.39i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.69 + 3.86i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.599 - 1.03i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.87iT - 53T^{2} \) |
| 59 | \( 1 + (6.18 - 10.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 2.05i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.69 - 3.86i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 3.37T + 73T^{2} \) |
| 79 | \( 1 + (8.55 - 4.94i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.82 + 6.61i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 11.9iT - 89T^{2} \) |
| 97 | \( 1 + (5.24 + 9.08i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−16.51765610552069472733345803342, −15.61075933047633612490793873657, −14.23924598717183196741207513412, −12.84199130253160756035144232492, −12.14967353072473799302919981303, −10.97513841663674793670155514433, −8.306669182248338593189779514203, −7.39524432508755506327575201218, −5.76614090183951599052206518532, −4.28240964674254658949224646414,
3.66052421143855954487685300223, 5.11295463965147680559141361569, 6.83566204816395454826821293633, 9.241917280934491138178181440648, 10.93046936472597146119361288247, 11.35169226933396810020936211550, 12.48218459409654697054743531020, 14.19277296849093172579569511062, 15.29339579780420801575202789694, 16.00858113642032640593457210810