L(s) = 1 | − 1.41i·2-s + 1.64i·3-s − 2.00·4-s + 4.64·5-s + 2.33·6-s + 4.36·7-s + 2.82i·8-s + 6.27·9-s − 6.56i·10-s + 5.73·11-s − 3.29i·12-s + 18.6i·13-s − 6.16i·14-s + 7.65i·15-s + 4.00·16-s + 6.59·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.549i·3-s − 0.500·4-s + 0.928·5-s + 0.388·6-s + 0.623·7-s + 0.353i·8-s + 0.697·9-s − 0.656i·10-s + 0.521·11-s − 0.274i·12-s + 1.43i·13-s − 0.440i·14-s + 0.510i·15-s + 0.250·16-s + 0.387·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.405982057\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.405982057\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 1.64iT - 9T^{2} \) |
| 5 | \( 1 - 4.64T + 25T^{2} \) |
| 7 | \( 1 - 4.36T + 49T^{2} \) |
| 11 | \( 1 - 5.73T + 121T^{2} \) |
| 13 | \( 1 - 18.6iT - 169T^{2} \) |
| 17 | \( 1 - 6.59T + 289T^{2} \) |
| 23 | \( 1 + 25.3T + 529T^{2} \) |
| 29 | \( 1 - 3.54iT - 841T^{2} \) |
| 31 | \( 1 + 29.6iT - 961T^{2} \) |
| 37 | \( 1 - 62.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 64.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 77.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 18.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 70.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 44.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 28.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 59.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 30.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 162.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 18.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 134. iT - 9.40e3T^{2} \) |
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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
−10.09359435220650823815132437845, −9.573604782324161494682332952972, −8.926115042921463718176305648118, −7.75002049080686878527270703721, −6.58761048346599132532167465452, −5.57643668457888339380797193432, −4.47643482699212880701116244791, −3.88132427973376483843694435089, −2.22498662173201565886747361185, −1.41446036047971390618383582767,
0.974586619920943700305591945918, 2.14925933789539914302453862383, 3.78109062466385344922050327274, 5.02458689032755007885735845746, 5.84809829574888247895977316106, 6.59082605883626979756852392424, 7.68553885921319569472740060333, 8.115549446782243099211249580243, 9.348526462608867582119336520299, 10.00985321417191881237375679110