L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 13-s + (−0.5 − 0.866i)19-s + 0.999·21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (−0.5 + 0.866i)39-s + 43-s + (−0.499 + 0.866i)49-s − 0.999·57-s + (−1 − 1.73i)61-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s − 13-s + (−0.5 − 0.866i)19-s + 0.999·21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)37-s + (−0.5 + 0.866i)39-s + 43-s + (−0.499 + 0.866i)49-s − 0.999·57-s + (−1 − 1.73i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9095334433\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9095334433\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 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 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 2T + T^{2} \) |
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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
−11.87470676436479390047875989284, −10.98824764380308101496714893490, −9.543962154888332755699143801123, −8.847230279027203571062064612439, −7.892333325174709491940246209738, −7.04936721592076829894613243241, −5.92539283495444043163189115436, −4.77245240657229405979929668072, −3.03372904482771312369732329605, −1.92725570838312540550105564611,
2.30826723556197991209654455590, 3.86627183256743128080161016582, 4.60711061314667166107247916241, 5.84514093538652224024396997107, 7.40669033595744115951681820005, 8.060402877826710628435774488488, 9.201412836095912561002994051191, 10.09505564991126101387692212510, 10.70505139185394982870972464412, 11.69777295426374333599805840855