L(s) = 1 | − 1.41i·2-s + 2.44·5-s − 2.82i·8-s − 3.46i·10-s − 1.41i·11-s + 5.19i·13-s − 4.00·16-s + 4.89·17-s − 1.73i·19-s − 2.00·22-s − 5.65i·23-s + 0.999·25-s + 7.34·26-s + 2.82i·29-s + 1.73i·31-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + 1.09·5-s − 0.999i·8-s − 1.09i·10-s − 0.426i·11-s + 1.44i·13-s − 1.00·16-s + 1.18·17-s − 0.397i·19-s − 0.426·22-s − 1.17i·23-s + 0.199·25-s + 1.44·26-s + 0.525i·29-s + 0.311i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36774 - 1.23956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36774 - 1.23956i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 2.82iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 - 7.07iT - 71T^{2} \) |
| 73 | \( 1 - 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 4.89T + 89T^{2} \) |
| 97 | \( 1 - 10.3iT - 97T^{2} \) |
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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
−10.91700869376417312906723322165, −10.04546534367812678095551293422, −9.495548447580230506320669642980, −8.482934848115219474816704761081, −6.95162811633682740881105368789, −6.29147651339654169986309928524, −5.02908713655860041858033502440, −3.66194779668353224752148952161, −2.46678554053813862554345726233, −1.42032966318660902566473438361,
1.82784642471843825147680567386, 3.25260139742650708884155216828, 5.20584833639502952382641569862, 5.65366490005888899785754801000, 6.58075505144410173824752364124, 7.67560926915879912079886185546, 8.255803617930082080140729742605, 9.666726365922565052133279120284, 10.13153090352513252765726337057, 11.28239593242971953768698872201