L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (1.22 + 0.707i)11-s − 0.999·16-s + (1.00 − 1.73i)22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + (1.22 − 0.707i)29-s + 1.41i·32-s + (−1.22 − 0.707i)44-s + (−1.00 − 1.73i)46-s + (1.22 + 0.707i)50-s + (1.22 − 0.707i)53-s + (−1.00 − 1.73i)58-s + 1.00·64-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (1.22 + 0.707i)11-s − 0.999·16-s + (1.00 − 1.73i)22-s + (1.22 − 0.707i)23-s + (−0.5 + 0.866i)25-s + (1.22 − 0.707i)29-s + 1.41i·32-s + (−1.22 − 0.707i)44-s + (−1.00 − 1.73i)46-s + (1.22 + 0.707i)50-s + (1.22 − 0.707i)53-s + (−1.00 − 1.73i)58-s + 1.00·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.414282318\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.414282318\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−8.848034635430777891516742261111, −7.71034077448195308635144420950, −6.85091479598227709712316069223, −6.32155869248754374303788272491, −5.05252172679609001610035991608, −4.34686779412842207822431867537, −3.64657663459570219428192199186, −2.78622285465543087952491659906, −1.88940739241199898980360440529, −0.989378628559122781933709126733,
1.19365289892052036394689059024, 2.64447363936065576326518963279, 3.71052475631998789304133246631, 4.57802350550996232162403728751, 5.35983666612812245471815012777, 6.12281662082391882242330802817, 6.63781732547682924717960052905, 7.28470120843508015059290113700, 8.034235230345414448244845674235, 8.864353684504348486508108029893