L(s) = 1 | + (1.40 + 0.139i)2-s + (−2.32 − 2.32i)3-s + (1.96 + 0.393i)4-s + (0.707 − 0.707i)5-s + (−2.94 − 3.59i)6-s + 0.982i·7-s + (2.70 + 0.828i)8-s + 7.82i·9-s + (1.09 − 0.896i)10-s + (−1.62 + 1.62i)11-s + (−3.64 − 5.47i)12-s + (−0.690 − 0.690i)13-s + (−0.137 + 1.38i)14-s − 3.28·15-s + (3.68 + 1.54i)16-s − 2.19·17-s + ⋯ |
L(s) = 1 | + (0.995 + 0.0989i)2-s + (−1.34 − 1.34i)3-s + (0.980 + 0.196i)4-s + (0.316 − 0.316i)5-s + (−1.20 − 1.46i)6-s + 0.371i·7-s + (0.956 + 0.292i)8-s + 2.60i·9-s + (0.345 − 0.283i)10-s + (−0.490 + 0.490i)11-s + (−1.05 − 1.58i)12-s + (−0.191 − 0.191i)13-s + (−0.0367 + 0.369i)14-s − 0.849·15-s + (0.922 + 0.386i)16-s − 0.532·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08283 - 0.450920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08283 - 0.450920i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.139i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (2.32 + 2.32i)T + 3iT^{2} \) |
| 7 | \( 1 - 0.982iT - 7T^{2} \) |
| 11 | \( 1 + (1.62 - 1.62i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.690 + 0.690i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 + (-1.92 - 1.92i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.01iT - 23T^{2} \) |
| 29 | \( 1 + (5.27 + 5.27i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.435T + 31T^{2} \) |
| 37 | \( 1 + (5.79 - 5.79i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.93iT - 41T^{2} \) |
| 43 | \( 1 + (0.507 - 0.507i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + (-6.29 + 6.29i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.67 - 5.67i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.60 + 3.60i)T + 61iT^{2} \) |
| 67 | \( 1 + (-4.53 - 4.53i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 9.24iT - 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + (0.683 + 0.683i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.44iT - 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
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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
−13.77716184960941211050112420459, −13.03298173956136939111099711811, −12.27502214178384176036045193140, −11.52577395144196121450121546647, −10.36058473490034610161954081099, −7.962036844478056711588190138421, −6.87963040737156692782462460964, −5.82776280358360560059081917157, −4.94254582513974706666841119238, −2.05849018613785828681745526692,
3.53145371421731720630799622830, 4.86884280850687598442586466180, 5.77409043113278268502091282870, 6.96658056239190043212446974780, 9.469543948878706704398319418765, 10.64853230540270707772625176351, 11.09502417809936256461583186745, 12.17533942547689782465379249828, 13.47087845790127091292922242231, 14.69956755543307108625693095900