L(s) = 1 | + 0.613i·3-s − 2.23·5-s − 2.64i·7-s + 2.62·9-s + 5.55i·11-s + 1.06·13-s − 1.37i·15-s + 5.75·17-s + 1.62·21-s + 5.00·25-s + 3.45i·27-s + 9.62·29-s − 3.40·33-s + 5.91i·35-s + 0.652i·39-s + ⋯ |
L(s) = 1 | + 0.354i·3-s − 0.999·5-s − 0.999i·7-s + 0.874·9-s + 1.67i·11-s + 0.294·13-s − 0.354i·15-s + 1.39·17-s + 0.354·21-s + 1.00·25-s + 0.664i·27-s + 1.78·29-s − 0.593·33-s + 0.999i·35-s + 0.104i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30201 + 0.348873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30201 + 0.348873i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 - 0.613iT - 3T^{2} \) |
| 11 | \( 1 - 5.55iT - 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 13.6iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 9.74iT - 79T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 19.3T + 97T^{2} \) |
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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
−10.50965637054285049082002159912, −10.21408707435206751624966487170, −9.241620464610096750370646220178, −7.82737319764693634131405630258, −7.46168695818839686105306739349, −6.54069702385118640783151437806, −4.76654256241646054048555899341, −4.33263776236038357050328031121, −3.26607130526066882779351456483, −1.26978788452515273596639357370,
1.00578099066002059224857071411, 2.88727573701469633190227850177, 3.82473895118450275634240307208, 5.17642469772371533749119447297, 6.14297972564421522364343202453, 7.13765517666647678191343009224, 8.282049449924275394743709900524, 8.516466304625432577133543295701, 9.855225860511856403542363586651, 10.79909581805685235819296693931