L(s) = 1 | − 1.73i·13-s + (1.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 43-s + (0.5 + 0.866i)67-s + (−1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯ |
L(s) = 1 | − 1.73i·13-s + (1.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + 43-s + (0.5 + 0.866i)67-s + (−1.5 + 0.866i)73-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)103-s + (−0.5 − 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.159088361\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.159088361\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 1.73iT - T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 - T^{2} \) |
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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
−9.687184356593749704493411338288, −8.403023092783124412015173049755, −7.972400335505062075757378460648, −7.18860862345448173196603323893, −6.02399397173807095499728119634, −5.53021260668522422814805144838, −4.49610482149762231237602058076, −3.41057191562855977404957052690, −2.60865273860038604852418654443, −1.04076825266703248020012614521,
1.39681928976638470793928278990, 2.61058847721582603051547490613, 3.70144622381404672599981968592, 4.62941671262291896795223466798, 5.42507612909214849578848355285, 6.49898595070475263277650220619, 7.11930821621898335104347898863, 7.891062340729175464212660120786, 9.081666323979300436048649978424, 9.275247870096338604079306453823