| L(s) = 1 | − i·2-s + 3.10i·3-s − 4-s + 3.10·6-s + i·7-s + i·8-s − 6.62·9-s + 11-s − 3.10i·12-s − 3.62i·13-s + 14-s + 16-s + 4.20i·17-s + 6.62i·18-s + 8.15·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.79i·3-s − 0.5·4-s + 1.26·6-s + 0.377i·7-s + 0.353i·8-s − 2.20·9-s + 0.301·11-s − 0.895i·12-s − 1.00i·13-s + 0.267·14-s + 0.250·16-s + 1.01i·17-s + 1.56i·18-s + 1.87·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.567601446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567601446\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 3.10iT - 3T^{2} \) |
| 13 | \( 1 + 3.62iT - 13T^{2} \) |
| 17 | \( 1 - 4.20iT - 17T^{2} \) |
| 19 | \( 1 - 8.15T + 19T^{2} \) |
| 23 | \( 1 - 0.897iT - 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 - 1.10iT - 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 - 1.15iT - 47T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 - 8.41iT - 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 0.205iT - 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 2.20iT - 83T^{2} \) |
| 89 | \( 1 - 7.45T + 89T^{2} \) |
| 97 | \( 1 + 2.25iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.065693054841288499462174152677, −8.283762564864226242280758153092, −7.55666225343519528610805700046, −5.95455624423959177112776398143, −5.64601914899165493518271416366, −4.68786511553076380657125872228, −4.17261726012155686453047691194, −3.11245301068131407123835866252, −2.90832360568776031304830938991, −1.17797781491862544732058209394,
0.53601677362706529827215153204, 1.39215980061542732184674910202, 2.49730751216123094343814103449, 3.48169899322836309179290777089, 4.69538622380400615613131919385, 5.51683007398141216139214170467, 6.31592505694000532951450997200, 6.90690470656334291181126800271, 7.44403492552324620392696710790, 7.81862209541781848854973435635
