L(s) = 1 | − 0.858i·2-s + (2.44 − 1.74i)3-s + 3.26·4-s + 2.23i·5-s + (−1.49 − 2.09i)6-s + 2.16·7-s − 6.23i·8-s + (2.92 − 8.51i)9-s + 1.92·10-s + 7.88i·11-s + (7.96 − 5.68i)12-s + 3.60·13-s − 1.86i·14-s + (3.89 + 5.45i)15-s + 7.69·16-s + 3.67i·17-s + ⋯ |
L(s) = 1 | − 0.429i·2-s + (0.813 − 0.581i)3-s + 0.815·4-s + 0.447i·5-s + (−0.249 − 0.349i)6-s + 0.309·7-s − 0.779i·8-s + (0.324 − 0.945i)9-s + 0.192·10-s + 0.716i·11-s + (0.663 − 0.473i)12-s + 0.277·13-s − 0.133i·14-s + (0.259 + 0.363i)15-s + 0.480·16-s + 0.216i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.11968 - 1.09115i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11968 - 1.09115i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.44 + 1.74i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 13 | \( 1 - 3.60T \) |
good | 2 | \( 1 + 0.858iT - 4T^{2} \) |
| 7 | \( 1 - 2.16T + 49T^{2} \) |
| 11 | \( 1 - 7.88iT - 121T^{2} \) |
| 17 | \( 1 - 3.67iT - 289T^{2} \) |
| 19 | \( 1 + 20.8T + 361T^{2} \) |
| 23 | \( 1 + 6.25iT - 529T^{2} \) |
| 29 | \( 1 - 2.29iT - 841T^{2} \) |
| 31 | \( 1 + 24.7T + 961T^{2} \) |
| 37 | \( 1 + 16.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 58.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6.89iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 34.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 78.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 98.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 67.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 66.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 138. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 34.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.08T + 9.40e3T^{2} \) |
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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
−12.28324761436236012601683046677, −11.14171605689644072061915999982, −10.31453509746488912829147133368, −9.157851283217594975143638168509, −7.928033477598226384157619620723, −7.07162118088066251238287711197, −6.16712002983077866636496836437, −4.06113847829528517377871042866, −2.73235763896436529434170761237, −1.66689319852118363570622666306,
2.01267010179038103839154068940, 3.48937432436440286417444228885, 4.94371735608356951347768140462, 6.15480175865972170497142259951, 7.53480839986193582714471873677, 8.357450476622968810489584249885, 9.204553992498658522443804978877, 10.57981953621688444148075218997, 11.20938673741593463566566747784, 12.48289989131823388558285564366