L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (−1.22 − 0.707i)14-s − 0.999·16-s − 1.41i·17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.22 − 0.707i)26-s + (−0.500 + 0.866i)28-s + (−1.22 + 0.707i)29-s + 1.41i·32-s − 2.00·34-s − 37-s + (1.22 − 0.707i)38-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (0.5 − 0.866i)7-s + (0.5 + 0.866i)13-s + (−1.22 − 0.707i)14-s − 0.999·16-s − 1.41i·17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.22 − 0.707i)26-s + (−0.500 + 0.866i)28-s + (−1.22 + 0.707i)29-s + 1.41i·32-s − 2.00·34-s − 37-s + (1.22 − 0.707i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.013067001\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.013067001\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.33121884750878190027237589250, −9.528250552858224149879384447391, −8.780642480251639825232732723821, −7.53763525882503913332849334949, −6.83834357517657678574077155501, −5.41793421500626672609231841332, −4.25664681685793625959237175840, −3.63902452148352536840972010628, −2.32749745492192979966623676654, −1.18548226312010315452344462970,
2.05562659794515010318026019945, 3.61297844467156653018053003630, 4.98049686532585446078343317032, 5.66032173504532130694895961784, 6.29707430781922614300828453286, 7.43238301724318775601834781827, 8.078751658927736438478630929204, 8.757790996548291017657043706957, 9.548716245772201586605556010637, 10.85948611738944536711513585816