L(s) = 1 | − 7.34·5-s + 10.3·7-s − 8.48·11-s − 10.3i·13-s + 21.2i·17-s + 20i·19-s − 14.6i·23-s + 29·25-s + 36.7·29-s − 51.9·31-s − 76.3·35-s + 41.5i·37-s + 72.1i·41-s + 40i·43-s + 73.4i·47-s + ⋯ |
L(s) = 1 | − 1.46·5-s + 1.48·7-s − 0.771·11-s − 0.799i·13-s + 1.24i·17-s + 1.05i·19-s − 0.638i·23-s + 1.15·25-s + 1.26·29-s − 1.67·31-s − 2.18·35-s + 1.12i·37-s + 1.75i·41-s + 0.930i·43-s + 1.56i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9672463988\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9672463988\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 7.34T + 25T^{2} \) |
| 7 | \( 1 - 10.3T + 49T^{2} \) |
| 11 | \( 1 + 8.48T + 121T^{2} \) |
| 13 | \( 1 + 10.3iT - 169T^{2} \) |
| 17 | \( 1 - 21.2iT - 289T^{2} \) |
| 19 | \( 1 - 20iT - 361T^{2} \) |
| 23 | \( 1 + 14.6iT - 529T^{2} \) |
| 29 | \( 1 - 36.7T + 841T^{2} \) |
| 31 | \( 1 + 51.9T + 961T^{2} \) |
| 37 | \( 1 - 41.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 72.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 73.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 100iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 73.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 20T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 127.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 40T + 9.40e3T^{2} \) |
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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
−10.89859065822478259742522223377, −10.17241560414313173577948749514, −8.552644415326185314859074624120, −7.949683514866878568690310594166, −7.72611572146458892032793100458, −6.18140374237505175978916644111, −4.96256319899987028533800689627, −4.24319550087343701880928052982, −3.06487555403443334989573372379, −1.37844627845838046588676763718,
0.39384308152121996320574497660, 2.16686380566576402469038978223, 3.64066597204061468515300603701, 4.67704728838389120694104637561, 5.28598076179657395638641023223, 7.18942127040742793820225738918, 7.43625859990946335420548614440, 8.464798937091910972471521899592, 9.129181189356564932660497156617, 10.61044576345497195111880272946