L(s) = 1 | + (2.41 − 0.647i)3-s + (−3.34 + 1.92i)7-s + (2.82 − 1.63i)9-s + (1.17 + 4.38i)11-s + (−2.93 + 2.09i)13-s + (−1.34 + 5.00i)17-s + (6.75 + 1.80i)19-s + (−6.82 + 6.82i)21-s + (−0.490 − 1.82i)23-s + (0.465 − 0.465i)27-s + (−1.71 − 0.990i)29-s + (−2.74 − 2.74i)31-s + (5.67 + 9.83i)33-s + (10.2 + 5.93i)37-s + (−5.72 + 6.97i)39-s + ⋯ |
L(s) = 1 | + (1.39 − 0.373i)3-s + (−1.26 + 0.729i)7-s + (0.941 − 0.543i)9-s + (0.354 + 1.32i)11-s + (−0.812 + 0.582i)13-s + (−0.325 + 1.21i)17-s + (1.54 + 0.415i)19-s + (−1.48 + 1.48i)21-s + (−0.102 − 0.381i)23-s + (0.0895 − 0.0895i)27-s + (−0.318 − 0.183i)29-s + (−0.493 − 0.493i)31-s + (0.988 + 1.71i)33-s + (1.69 + 0.976i)37-s + (−0.916 + 1.11i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.057201921\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.057201921\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.93 - 2.09i)T \) |
good | 3 | \( 1 + (-2.41 + 0.647i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (3.34 - 1.92i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 4.38i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (1.34 - 5.00i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-6.75 - 1.80i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.490 + 1.82i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.71 + 0.990i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.74 + 2.74i)T + 31iT^{2} \) |
| 37 | \( 1 + (-10.2 - 5.93i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.04 + 1.88i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (5.07 + 1.35i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + 2.88iT - 47T^{2} \) |
| 53 | \( 1 + (3.37 + 3.37i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.22 - 4.58i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-1.59 - 2.76i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.60 + 2.78i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.70 + 6.36i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + 3.35T + 73T^{2} \) |
| 79 | \( 1 - 0.191iT - 79T^{2} \) |
| 83 | \( 1 - 7.50iT - 83T^{2} \) |
| 89 | \( 1 + (-10.5 + 2.83i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (3.13 + 5.43i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.506943937802916675664973220496, −9.236366380705785581117347198845, −8.175944564942966406346030293666, −7.43801975563436485548157301524, −6.72495444104556048370569084724, −5.80145498488642678756822408503, −4.43954910520057956703176659305, −3.49786301424103595123957446496, −2.58266523999544128311542659668, −1.80752743042546095409127455800,
0.70178025254273693930564547446, 2.74465242212124558467383004647, 3.16740253327952246532051612434, 3.94870556059209226129832544480, 5.19134612991364774857721886222, 6.28028967873150379353057326499, 7.37081919445338128945729735311, 7.75694436133778622373811486211, 8.969395244789270911641562120799, 9.457415592647694386783565234036