L(s) = 1 | + (0.5 − 0.866i)7-s + (1.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (1 + 1.73i)37-s + 43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 − 0.866i)73-s + (−1 − 1.73i)79-s + 1.73i·97-s + (−0.5 + 0.866i)109-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)7-s + (1.5 − 0.866i)19-s + (−0.5 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (1 + 1.73i)37-s + 43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (−1.5 − 0.866i)73-s + (−1 − 1.73i)79-s + 1.73i·97-s + (−0.5 + 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.017409161\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.017409161\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−10.52763465050665019733674135406, −9.640117839765748790219414294167, −8.897696436817363988728595590791, −7.57121436215882822954182434071, −7.39122724278161595238615136385, −6.04622910470620979602235280797, −5.04931520994212375136737243929, −4.10737781036924716793337064531, −2.96844602323825526379510974099, −1.34901770569908910157720622206,
1.72959302221823731376959918275, 2.99628300238794761710597513424, 4.22198973540775874696625740554, 5.41675416550233053352327716090, 5.95869670364095433598890288696, 7.32121848030643215348758627346, 7.981470997200612890847591784955, 8.993744850455697861459664885527, 9.609291742336016477735603166610, 10.68102768420910412258500959013