L(s) = 1 | − 0.722i·2-s + 3.47·4-s − 7.94i·5-s + 2.64·7-s − 5.40i·8-s − 5.74·10-s + 9.25i·11-s − 22.1·13-s − 1.91i·14-s + 10.0·16-s − 27.2i·17-s + 31.8·19-s − 27.6i·20-s + 6.68·22-s + 13.5i·23-s + ⋯ |
L(s) = 1 | − 0.361i·2-s + 0.869·4-s − 1.58i·5-s + 0.377·7-s − 0.675i·8-s − 0.574·10-s + 0.841i·11-s − 1.70·13-s − 0.136i·14-s + 0.625·16-s − 1.60i·17-s + 1.67·19-s − 1.38i·20-s + 0.303·22-s + 0.588i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.27140 - 1.27140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27140 - 1.27140i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 2 | \( 1 + 0.722iT - 4T^{2} \) |
| 5 | \( 1 + 7.94iT - 25T^{2} \) |
| 11 | \( 1 - 9.25iT - 121T^{2} \) |
| 13 | \( 1 + 22.1T + 169T^{2} \) |
| 17 | \( 1 + 27.2iT - 289T^{2} \) |
| 19 | \( 1 - 31.8T + 361T^{2} \) |
| 23 | \( 1 - 13.5iT - 529T^{2} \) |
| 29 | \( 1 + 6.73iT - 841T^{2} \) |
| 31 | \( 1 - 11.9T + 961T^{2} \) |
| 37 | \( 1 - 26.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 63.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 0.518iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 71.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 38.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 27.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 29.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 67.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 18.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 80.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 62.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 46.7T + 9.40e3T^{2} \) |
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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
−11.96113079715228265520056808970, −11.56831763677089310601634412029, −9.755463062016051583295097726129, −9.527098965185142540390871717709, −7.82518396594787692474883933165, −7.18701076991642243079408407702, −5.35700457484492641854227630263, −4.62510932171510529676916049330, −2.63859987119320154898442511320, −1.12814751984459438724577775041,
2.28398247401884222969156153547, 3.37459480461844376771359875881, 5.42540274591392708659735991127, 6.50826297022382866825580297001, 7.31695713159108757195373472276, 8.142812062235038470983888240953, 9.915012208670637069363109787213, 10.70517841607096667671420666049, 11.44643151638168686401382278405, 12.32876447158557714889027752047