L(s) = 1 | + 3.77i·2-s + (2.45 + 1.72i)3-s − 10.2·4-s + 1.62i·5-s + (−6.52 + 9.27i)6-s − 7.78·7-s − 23.7i·8-s + (3.04 + 8.46i)9-s − 6.15·10-s − 2.72i·11-s + (−25.2 − 17.7i)12-s + 9.61·13-s − 29.4i·14-s + (−2.80 + 3.99i)15-s + 48.6·16-s + 27.3i·17-s + ⋯ |
L(s) = 1 | + 1.88i·2-s + (0.818 + 0.575i)3-s − 2.57·4-s + 0.325i·5-s + (−1.08 + 1.54i)6-s − 1.11·7-s − 2.97i·8-s + (0.338 + 0.940i)9-s − 0.615·10-s − 0.248i·11-s + (−2.10 − 1.47i)12-s + 0.739·13-s − 2.10i·14-s + (−0.187 + 0.266i)15-s + 3.04·16-s + 1.61i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.577749 - 1.11235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.577749 - 1.11235i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.45 - 1.72i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 - 3.77iT - 4T^{2} \) |
| 5 | \( 1 - 1.62iT - 25T^{2} \) |
| 7 | \( 1 + 7.78T + 49T^{2} \) |
| 11 | \( 1 + 2.72iT - 121T^{2} \) |
| 13 | \( 1 - 9.61T + 169T^{2} \) |
| 17 | \( 1 - 27.3iT - 289T^{2} \) |
| 19 | \( 1 + 27.7T + 361T^{2} \) |
| 23 | \( 1 - 3.53iT - 529T^{2} \) |
| 29 | \( 1 - 6.30iT - 841T^{2} \) |
| 31 | \( 1 - 39.9T + 961T^{2} \) |
| 37 | \( 1 - 44.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 50.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 47.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 9.97iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 71.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 86.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 119. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 125. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 137. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 0.0640T + 9.40e3T^{2} \) |
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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
−13.27373058825400426359674671298, −12.97056197330445969773512413806, −10.58962573817458119216673718558, −9.727565465766888087350147401908, −8.627015357334832068292630471145, −8.158303096236679925683259165208, −6.68814743361468796804057592407, −6.08926360423633249167102409264, −4.49188397446461942470524143748, −3.43990278632188987144493992578,
0.71348416701859677728474483483, 2.38202084120453341056919212713, 3.34065175021626594850917219128, 4.56291520717395643890352281652, 6.52901162948535750108760593428, 8.234246454950580632689928692015, 9.155710818625517462201212403396, 9.746898104967779970808158337094, 10.87453532093562527190949297259, 12.06163689449325587717725331978