L(s) = 1 | + (−0.982 − 0.183i)3-s + (0.739 + 0.673i)4-s + (−0.876 + 1.75i)7-s + (0.932 + 0.361i)9-s + (−0.602 − 0.798i)12-s + (0.658 − 1.32i)13-s + (0.0922 + 0.995i)16-s + (0.538 − 0.100i)19-s + (1.18 − 1.56i)21-s + (−0.850 + 0.526i)25-s + (−0.850 − 0.526i)27-s + (−1.83 + 0.710i)28-s + (−0.111 − 1.20i)31-s + (0.445 + 0.895i)36-s + (−0.537 − 0.711i)37-s + ⋯ |
L(s) = 1 | + (−0.982 − 0.183i)3-s + (0.739 + 0.673i)4-s + (−0.876 + 1.75i)7-s + (0.932 + 0.361i)9-s + (−0.602 − 0.798i)12-s + (0.658 − 1.32i)13-s + (0.0922 + 0.995i)16-s + (0.538 − 0.100i)19-s + (1.18 − 1.56i)21-s + (−0.850 + 0.526i)25-s + (−0.850 − 0.526i)27-s + (−1.83 + 0.710i)28-s + (−0.111 − 1.20i)31-s + (0.445 + 0.895i)36-s + (−0.537 − 0.711i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6527360066\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6527360066\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.982 + 0.183i)T \) |
| 103 | \( 1 + (0.602 + 0.798i)T \) |
good | 2 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 5 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 7 | \( 1 + (0.876 - 1.75i)T + (-0.602 - 0.798i)T^{2} \) |
| 11 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 13 | \( 1 + (-0.658 + 1.32i)T + (-0.602 - 0.798i)T^{2} \) |
| 17 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 19 | \( 1 + (-0.538 + 0.100i)T + (0.932 - 0.361i)T^{2} \) |
| 23 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 29 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 31 | \( 1 + (0.111 + 1.20i)T + (-0.982 + 0.183i)T^{2} \) |
| 37 | \( 1 + (0.537 + 0.711i)T + (-0.273 + 0.961i)T^{2} \) |
| 41 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 43 | \( 1 + (-1.02 + 1.35i)T + (-0.273 - 0.961i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 59 | \( 1 + (0.602 - 0.798i)T^{2} \) |
| 61 | \( 1 + (0.156 - 0.0971i)T + (0.445 - 0.895i)T^{2} \) |
| 67 | \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \) |
| 71 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 73 | \( 1 + (0.0505 - 0.177i)T + (-0.850 - 0.526i)T^{2} \) |
| 79 | \( 1 + (0.243 + 0.857i)T + (-0.850 + 0.526i)T^{2} \) |
| 83 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 89 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 97 | \( 1 + (-1.44 - 0.895i)T + (0.445 + 0.895i)T^{2} \) |
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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
−12.00934465976821835938444142689, −11.39650784249685318237350542462, −10.37930040367435353343384624222, −9.258294141155660654337814370873, −8.134725305935256304220646476431, −7.10543591083731645883960277442, −5.90797008390197426625116375172, −5.61141753265880979232118929251, −3.57728933578770016329019917682, −2.32506086333377622741730086950,
1.30049452879220619474988422114, 3.62898575853660842174255863207, 4.71301008518075433565194025353, 6.18335867114909723218627050021, 6.69020253633633425579078356808, 7.50516487887005785325679978737, 9.460799673304488691631842931966, 10.12914428895467493610928154178, 10.85957664525650653306845386004, 11.51681524721732762161117173811