L(s) = 1 | − 3-s + 5-s + (2.01 − 3.48i)7-s + 9-s + (1.47 − 2.56i)11-s + (−0.795 − 1.37i)13-s − 15-s + (−2.62 − 4.54i)17-s + (0.339 + 0.587i)19-s + (−2.01 + 3.48i)21-s + (−0.358 − 0.620i)23-s + 25-s − 27-s + (3.40 − 5.90i)29-s + (1.61 − 2.78i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + (0.761 − 1.31i)7-s + 0.333·9-s + (0.445 − 0.772i)11-s + (−0.220 − 0.382i)13-s − 0.258·15-s + (−0.636 − 1.10i)17-s + (0.0778 + 0.134i)19-s + (−0.439 + 0.761i)21-s + (−0.0747 − 0.129i)23-s + 0.200·25-s − 0.192·27-s + (0.632 − 1.09i)29-s + (0.289 − 0.500i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.459353467\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.459353467\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.98 - 7.62i)T \) |
good | 7 | \( 1 + (-2.01 + 3.48i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.47 + 2.56i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.795 + 1.37i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.62 + 4.54i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.339 - 0.587i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.358 + 0.620i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.40 + 5.90i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.61 + 2.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.94 + 3.37i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.662 - 1.14i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + (3.03 - 5.26i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.84T + 53T^{2} \) |
| 59 | \( 1 - 3.21T + 59T^{2} \) |
| 61 | \( 1 + (-0.514 - 0.890i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (6.97 - 12.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.44 + 5.96i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.19 + 8.99i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.85 - 6.68i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 0.969T + 89T^{2} \) |
| 97 | \( 1 + (-4.77 - 8.26i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.069544045829136408953991076363, −7.38357148110055798640959838045, −6.68441561718860285307863919444, −6.00922297924531878680213772049, −5.07748838870412033134113451350, −4.50743720478396928103303738612, −3.67707563360051303883159501388, −2.53283519898041596586116145594, −1.31071427810578543167039692682, −0.46098479019010255791456453202,
1.59656304348831158282694346180, 2.00901533297073767895410798555, 3.24059251767992937219882749184, 4.49365254336365221611431008645, 4.99334629812139658903699287869, 5.70656305119238006440862464690, 6.53827590270968056441955744012, 6.98876927913309970679838252738, 8.202106758925625268653405659956, 8.681411222976241486491317292243