L(s) = 1 | + i·2-s + 1.79i·3-s − 4-s − 1.79·6-s − 2.79i·7-s − i·8-s − 0.208·9-s + 3.79·11-s − 1.79i·12-s + 1.20i·13-s + 2.79·14-s + 16-s + 3.79i·17-s − 0.208i·18-s − 1.20·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.03i·3-s − 0.5·4-s − 0.731·6-s − 1.05i·7-s − 0.353i·8-s − 0.0695·9-s + 1.14·11-s − 0.517i·12-s + 0.335i·13-s + 0.746·14-s + 0.250·16-s + 0.919i·17-s − 0.0491i·18-s − 0.277·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.660161901\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.660161901\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 1.79iT - 3T^{2} \) |
| 7 | \( 1 + 2.79iT - 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 - 1.20iT - 13T^{2} \) |
| 17 | \( 1 - 3.79iT - 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 29 | \( 1 - 1.58T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 + 7.16iT - 43T^{2} \) |
| 47 | \( 1 - 13.5iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 - 7.16iT - 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 + 14.7iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 3.16T + 89T^{2} \) |
| 97 | \( 1 + 14.9iT - 97T^{2} \) |
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
−10.09922298474026099650807761649, −9.229867233457764723539480574501, −8.532002671330897523529697683309, −7.51835413098945120501715516639, −6.69246874172419418028203814061, −5.97454582131443774795944625801, −4.54224110811281379613813774736, −4.30519325339976193162150469409, −3.36338250020110385661381208095, −1.28303242384288746874006320830,
0.877650825268189882388011658252, 2.03912123681488641838359439106, 2.89234477993705233359980567544, 4.18500592664710166601502314489, 5.25400807101062211199822904272, 6.30330124635046198448482594238, 6.93770323003207233789685972182, 8.071223579601287973234763166451, 8.711167209673644749909748647696, 9.542115031487621164532454885509