L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 0.999·55-s + (0.5 − 0.866i)61-s + (0.5 + 0.866i)73-s + (−0.499 − 0.866i)81-s + 2·83-s + 0.999·85-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (0.499 + 0.866i)45-s + (0.5 − 0.866i)47-s + 0.999·55-s + (0.5 − 0.866i)61-s + (0.5 + 0.866i)73-s + (−0.499 − 0.866i)81-s + 2·83-s + 0.999·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.263252575\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.263252575\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 2T + T^{2} \) |
| 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
−8.717109917227330457579181591885, −8.124982455574306957142741798460, −7.50783802004575073445301186780, −6.48957887246313796201305956739, −5.61881413269342878491488954252, −5.23116276040291331517536910185, −4.23782174999867611170028495022, −3.47791715399893294117677873613, −2.01209835935887637317845673145, −1.56922280159680543184160473022,
0.72593903728398656606338243250, 2.34453309674846738681748061847, 2.97264276023836715606033146379, 3.81565440144836153143760433219, 4.81419628247920123479889106032, 5.85600528650101313618957763664, 6.43768228897130868959087639998, 6.81166801410978967905108111328, 7.909236072443285415643004601586, 8.699629564000920235757333711119