L(s) = 1 | + (−1.5 + 0.866i)2-s + 1.73i·3-s + (0.5 − 0.866i)4-s + (−1.49 − 2.59i)6-s + (2.5 − 0.866i)7-s − 1.73i·8-s − 2.99·9-s + (3 + 1.73i)11-s + (1.50 + 0.866i)12-s + 3.46i·13-s + (−3 + 3.46i)14-s + (2.49 + 4.33i)16-s + (−3 + 5.19i)17-s + (4.49 − 2.59i)18-s + (−6 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (−1.06 + 0.612i)2-s + 0.999i·3-s + (0.250 − 0.433i)4-s + (−0.612 − 1.06i)6-s + (0.944 − 0.327i)7-s − 0.612i·8-s − 0.999·9-s + (0.904 + 0.522i)11-s + (0.433 + 0.249i)12-s + 0.960i·13-s + (−0.801 + 0.925i)14-s + (0.624 + 1.08i)16-s + (−0.727 + 1.26i)17-s + (1.06 − 0.612i)18-s + (−1.37 + 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0219586 + 0.692742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0219586 + 0.692742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 2 | \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6 - 3.46i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 0.866i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.92iT - 71T^{2} \) |
| 73 | \( 1 + (3 + 1.73i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−10.90804375699040920097146219675, −10.27271836141236963311890415781, −9.328328095508950100191348210440, −8.676847126394447138055225033601, −8.046579033712958228665655895707, −6.84651061662732352503594710611, −6.02477940787102558058016703567, −4.34381341930807007258902733836, −4.04685548189866241618608465926, −1.77679179106632974928780606415,
0.59211553365999991701221482196, 1.85645975410394283444172087028, 2.87272187373347892593797480937, 4.83966341408845863366616686658, 5.89620916215843162358149701948, 7.07826082841924579101631793550, 7.975471045820858145695861668465, 8.810968646450715539072660421692, 9.181332958396641308130262522563, 10.72488767366802184456444912170