L(s) = 1 | + 0.414i·2-s + 1.82·4-s − 2.82i·7-s + 1.58i·8-s + 2·11-s − i·13-s + 1.17·14-s + 3·16-s + 7.65i·17-s + 2.82·19-s + 0.828i·22-s + 4i·23-s + 0.414·26-s − 5.17i·28-s + 2·29-s + ⋯ |
L(s) = 1 | + 0.292i·2-s + 0.914·4-s − 1.06i·7-s + 0.560i·8-s + 0.603·11-s − 0.277i·13-s + 0.313·14-s + 0.750·16-s + 1.85i·17-s + 0.648·19-s + 0.176i·22-s + 0.834i·23-s + 0.0812·26-s − 0.977i·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.567414704\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.567414704\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 7.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 7.65iT - 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 1.65iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 6.82iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 0.343iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 3.65iT - 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 3.65iT - 97T^{2} \) |
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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
−8.508388459134827983871735447866, −8.027632820303162828625970683374, −7.16060157252890108062250857843, −6.68724875475662664340033751251, −5.91001534573219021824811310562, −5.08152149854055898974082981066, −3.85300852874425485903479155126, −3.39255877958607424364648779652, −1.99418483269219665828802028232, −1.12940611922741700079296770363,
0.943020994419852943220650855574, 2.24132920146476728134256335831, 2.76138126953161816048022042855, 3.75404560435893750640614808880, 4.93030489900748785348192935870, 5.65457677001859684880475237834, 6.52915409288721407317265713176, 7.07795745040335573044612733718, 7.88448245392699650906278725017, 8.878925045255226430507034036808