L(s) = 1 | − 3-s − 5-s + (1.82 + 3.15i)7-s + 9-s + (1.30 + 2.26i)11-s + (1.63 − 2.82i)13-s + 15-s + (2.07 − 3.59i)17-s + (0.703 − 1.21i)19-s + (−1.82 − 3.15i)21-s + (0.502 − 0.870i)23-s + 25-s − 27-s + (−1.80 − 3.12i)29-s + (−3.41 − 5.91i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + (0.688 + 1.19i)7-s + 0.333·9-s + (0.393 + 0.681i)11-s + (0.452 − 0.784i)13-s + 0.258·15-s + (0.503 − 0.871i)17-s + (0.161 − 0.279i)19-s + (−0.397 − 0.688i)21-s + (0.104 − 0.181i)23-s + 0.200·25-s − 0.192·27-s + (−0.335 − 0.580i)29-s + (−0.613 − 1.06i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.542702771\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542702771\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (5.76 + 5.81i)T \) |
good | 7 | \( 1 + (-1.82 - 3.15i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.30 - 2.26i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.63 + 2.82i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.703 + 1.21i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.502 + 0.870i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.80 + 3.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.41 + 5.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.80 + 4.86i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.37 - 2.37i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 + (-1.31 - 2.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + (-7.36 + 12.7i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (1.79 + 3.11i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.56 + 4.43i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.86 - 11.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.72 - 13.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + (3.74 - 6.48i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.220649900533028723499384224619, −7.76367936655641977481661331293, −6.97515979831605243410528987739, −6.00660973812873779671148218538, −5.45198524266098246271309438475, −4.76855032408000733401888775666, −3.92044102051642504699860530218, −2.81769861839928334214072627553, −1.89880833667879276453658874642, −0.62087365996971544088658017907,
0.967290113363420507390912674048, 1.62746958558244721770788913488, 3.33287814754164887801394720329, 3.96826880465696343869451024561, 4.59600827353614749785369908814, 5.55407999600938442208491109067, 6.28743344890418733000454276019, 7.13392725908736621130839117153, 7.58709924544641874045964300808, 8.495859807927250177171259885195