L(s) = 1 | + (−0.453 + 0.891i)2-s + (0.309 + 0.951i)3-s + (−0.587 − 0.809i)4-s + (−0.987 − 0.156i)6-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)9-s + (0.243 + 0.398i)11-s + (0.587 − 0.809i)12-s + (−0.309 + 0.951i)16-s + (−0.178 + 0.744i)17-s + (−0.156 − 0.987i)18-s + (0.156 + 1.98i)19-s + (−0.465 + 0.0366i)22-s + (0.453 + 0.891i)24-s + (−0.951 − 0.309i)25-s + ⋯ |
L(s) = 1 | + (−0.453 + 0.891i)2-s + (0.309 + 0.951i)3-s + (−0.587 − 0.809i)4-s + (−0.987 − 0.156i)6-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)9-s + (0.243 + 0.398i)11-s + (0.587 − 0.809i)12-s + (−0.309 + 0.951i)16-s + (−0.178 + 0.744i)17-s + (−0.156 − 0.987i)18-s + (0.156 + 1.98i)19-s + (−0.465 + 0.0366i)22-s + (0.453 + 0.891i)24-s + (−0.951 − 0.309i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7813324669\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7813324669\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.453 - 0.891i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.156 + 0.987i)T \) |
good | 5 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 + (0.156 + 0.987i)T^{2} \) |
| 11 | \( 1 + (-0.243 - 0.398i)T + (-0.453 + 0.891i)T^{2} \) |
| 13 | \( 1 + (0.987 + 0.156i)T^{2} \) |
| 17 | \( 1 + (0.178 - 0.744i)T + (-0.891 - 0.453i)T^{2} \) |
| 19 | \( 1 + (-0.156 - 1.98i)T + (-0.987 + 0.156i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (0.891 - 0.453i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-1.69 - 0.863i)T + (0.587 + 0.809i)T^{2} \) |
| 47 | \( 1 + (-0.156 + 0.987i)T^{2} \) |
| 53 | \( 1 + (-0.891 + 0.453i)T^{2} \) |
| 59 | \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 67 | \( 1 + (0.546 - 0.891i)T + (-0.453 - 0.891i)T^{2} \) |
| 71 | \( 1 + (-0.453 + 0.891i)T^{2} \) |
| 73 | \( 1 + (-0.437 + 0.437i)T - iT^{2} \) |
| 79 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + 1.61iT - T^{2} \) |
| 89 | \( 1 + (1.29 + 1.10i)T + (0.156 + 0.987i)T^{2} \) |
| 97 | \( 1 + (0.133 + 0.0819i)T + (0.453 + 0.891i)T^{2} \) |
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
−10.13927027346640801983080172359, −9.764554749086130906179758744987, −8.798013287826193926385408075965, −8.144135853956800977112706271174, −7.39565770213875950803598976331, −6.10631349747722771828369458715, −5.57053822079153323678944839653, −4.37378185796687587781450717186, −3.72505838923294495172294020406, −1.92673301335269606055719707329,
0.873926974778355221834369820291, 2.30663092180870447473393523714, 3.06102894412905703312115605950, 4.28040025314901041648342010859, 5.50775899219876150600973274158, 6.79126532027886940585537525623, 7.44985024576350242121841741602, 8.324156865466653980407701407971, 9.139133262131429339809497411420, 9.592491684510101583554305059589