L(s) = 1 | + 2i·2-s − 2·4-s + 3i·5-s + i·7-s − 6·10-s + (2 + 3i)13-s − 2·14-s − 4·16-s + 2·17-s − i·19-s − 6i·20-s + 23-s − 4·25-s + (−6 + 4i)26-s − 2i·28-s − 5·29-s + ⋯ |
L(s) = 1 | + 1.41i·2-s − 4-s + 1.34i·5-s + 0.377i·7-s − 1.89·10-s + (0.554 + 0.832i)13-s − 0.534·14-s − 16-s + 0.485·17-s − 0.229i·19-s − 1.34i·20-s + 0.208·23-s − 0.800·25-s + (−1.17 + 0.784i)26-s − 0.377i·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.402449 - 1.32920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.402449 - 1.32920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-2 - 3i)T \) |
good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 5 | \( 1 - 3iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 9iT - 89T^{2} \) |
| 97 | \( 1 - 13iT - 97T^{2} \) |
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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
−10.86886749961792428436370057998, −9.611194872241140155710546806198, −8.904367246599325213759210513815, −7.83092885676215183737260321766, −7.25526601724211495470022035448, −6.37652777732491049683915618624, −5.93215685579438675684257100549, −4.73694598141883719712751855191, −3.47734682414405739433088248348, −2.22725512495290528709750834833,
0.71299461444317999534864209824, 1.64514815958258871066860468702, 3.11879311128771217160959458667, 4.02936398096495684560903501212, 4.95002915943273554773508624904, 5.93630370935974896079491771885, 7.36806488411198938482497420710, 8.349483547880858567401818051772, 9.137505264505083486332439864372, 9.829263423240537081908856520972