| L(s) = 1 | + (−0.629 − 1.93i)3-s + (−1.14 − 0.828i)5-s + (2.51 + 0.817i)7-s + (3.92 − 2.85i)9-s + (−9.49 − 5.55i)11-s + (−4.49 − 6.18i)13-s + (−0.886 + 2.72i)15-s + (1.70 − 2.34i)17-s + (6.60 − 2.14i)19-s − 5.38i·21-s − 13.6·23-s + (−7.11 − 21.8i)25-s + (−22.8 − 16.5i)27-s + (−48.1 − 15.6i)29-s + (−13.0 + 9.45i)31-s + ⋯ |
| L(s) = 1 | + (−0.209 − 0.645i)3-s + (−0.228 − 0.165i)5-s + (0.359 + 0.116i)7-s + (0.436 − 0.317i)9-s + (−0.862 − 0.505i)11-s + (−0.345 − 0.475i)13-s + (−0.0591 + 0.181i)15-s + (0.100 − 0.138i)17-s + (0.347 − 0.113i)19-s − 0.256i·21-s − 0.592·23-s + (−0.284 − 0.875i)25-s + (−0.845 − 0.614i)27-s + (−1.66 − 0.539i)29-s + (−0.419 + 0.304i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.408629 - 0.971535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.408629 - 0.971535i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.51 - 0.817i)T \) |
| 11 | \( 1 + (9.49 + 5.55i)T \) |
| good | 3 | \( 1 + (0.629 + 1.93i)T + (-7.28 + 5.29i)T^{2} \) |
| 5 | \( 1 + (1.14 + 0.828i)T + (7.72 + 23.7i)T^{2} \) |
| 13 | \( 1 + (4.49 + 6.18i)T + (-52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 2.34i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (-6.60 + 2.14i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + 13.6T + 529T^{2} \) |
| 29 | \( 1 + (48.1 + 15.6i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (13.0 - 9.45i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-5.47 + 16.8i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-29.9 + 9.72i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + 5.54iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-2.58 - 7.95i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-17.5 + 12.7i)T + (868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (16.4 - 50.7i)T + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-58.4 + 80.4i)T + (-1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 2.43T + 4.48e3T^{2} \) |
| 71 | \( 1 + (73.0 + 53.0i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-84.2 - 27.3i)T + (4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-70.1 - 96.4i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-33.4 + 46.0i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 - 146.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-66.3 + 48.2i)T + (2.90e3 - 8.94e3i)T^{2} \) |
| 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
−11.26355716191782540499690205298, −10.26506968403236748317518858928, −9.232715221866640655544564333894, −7.958054565441236343031717676997, −7.47477065694296821282115779118, −6.15505397062532688334586508084, −5.20965403269418376425230996696, −3.81365533301094098839704756063, −2.20603000953972500155068655024, −0.50882154627769508981097035613,
1.96930135103316820414784844014, 3.69835057330347137447679779996, 4.74068579336072868021223829527, 5.63597168973859564388937088003, 7.24770403594024358611680431033, 7.79688465843255426945017629360, 9.239787037537599677806312829167, 10.02384899512430908654254546107, 10.85930054560489110034161296197, 11.60729009930266007880812667472
