L(s) = 1 | + (0.395 − 0.684i)2-s + (4.5 + 7.79i)3-s + (15.6 + 27.1i)4-s + (−52.0 + 90.2i)5-s + 7.11·6-s + 50.1·8-s + (−40.5 + 70.1i)9-s + (41.1 + 71.3i)10-s + (248. + 430. i)11-s + (−141. + 244. i)12-s + 206.·13-s − 937.·15-s + (−482. + 835. i)16-s + (31.5 + 54.6i)17-s + (32.0 + 55.4i)18-s + (661. − 1.14e3i)19-s + ⋯ |
L(s) = 1 | + (0.0698 − 0.121i)2-s + (0.288 + 0.499i)3-s + (0.490 + 0.849i)4-s + (−0.931 + 1.61i)5-s + 0.0807·6-s + 0.276·8-s + (−0.166 + 0.288i)9-s + (0.130 + 0.225i)10-s + (0.620 + 1.07i)11-s + (−0.283 + 0.490i)12-s + 0.338·13-s − 1.07·15-s + (−0.470 + 0.815i)16-s + (0.0265 + 0.0459i)17-s + (0.0232 + 0.0403i)18-s + (0.420 − 0.728i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.867406822\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.867406822\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.395 + 0.684i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (52.0 - 90.2i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-248. - 430. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 206.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-31.5 - 54.6i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-661. + 1.14e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-97.2 + 168. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 4.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (3.76e3 + 6.51e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (5.17e3 - 8.96e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 4.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.19e3 - 3.80e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (8.89e3 + 1.54e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.75e3 - 3.03e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (5.31e3 - 9.20e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-6.63e3 - 1.14e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.56e4 - 2.71e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.97e4 - 3.42e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (5.64e4 - 9.77e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 3.03e4T + 8.58e9T^{2} \) |
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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
−12.34338230244581497594595244782, −11.47909389830708197241721965089, −10.85194713320570252959714424415, −9.716120420431017603440657761599, −8.233168868560332888258640586218, −7.29185791633888445297380321127, −6.58840893187143643147768595301, −4.31752645131620556006845878574, −3.42385713808556408189656861097, −2.41655969477833661649222774965,
0.62445258124539311716451783425, 1.46994270815991637902921007129, 3.61824299976399470164633089295, 5.05307791406769457207849168895, 6.10877602407146541108154319102, 7.48700538365522744423094121686, 8.529685605599663516265393905571, 9.271989989653898454165823910815, 10.84726101585602912697721895784, 11.83343391490364981000777639055