L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + (−2.29 + 3.96i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.563 + 0.325i)11-s + 0.999i·12-s + (−3.35 − 1.31i)13-s + 4.58·14-s + (−0.5 − 0.866i)16-s + (3.03 + 1.75i)17-s − 0.999·18-s + (−2.50 − 1.44i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.353 − 0.204i)6-s + (−0.865 + 1.49i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.169 + 0.0981i)11-s + 0.288i·12-s + (−0.931 − 0.364i)13-s + 1.22·14-s + (−0.125 − 0.216i)16-s + (0.735 + 0.424i)17-s − 0.235·18-s + (−0.575 − 0.332i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3692146797\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3692146797\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.35 + 1.31i)T \) |
good | 7 | \( 1 + (2.29 - 3.96i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.563 - 0.325i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.03 - 1.75i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.50 + 1.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.46 + 1.42i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.78 + 8.28i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.58iT - 31T^{2} \) |
| 37 | \( 1 + (-2.31 - 4.00i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.03 - 1.75i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.68 + 3.85i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.08T + 47T^{2} \) |
| 53 | \( 1 - 5.58iT - 53T^{2} \) |
| 59 | \( 1 + (6.47 + 3.73i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.31 + 5.74i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.58 + 7.93i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.2 + 6.51i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 1.04T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + (-7.10 + 4.09i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.21 - 12.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.906483940558432952314828876471, −8.180940894858319785005233481928, −7.49372122911473793392329918692, −6.40386249882381934630755645106, −5.68231380194977135980534709494, −4.61382512956884284291249965790, −3.37646668912167881161226250527, −2.66319147493757764564942638787, −1.96386332955484509218552224669, −0.13870271607291597194643740250,
1.37722934379442543461538216658, 2.97847724938441786722201405050, 3.79474575419592992252279820739, 4.69988177652612543698352353850, 5.58134806846802711416832619004, 6.84251684489559602893606055056, 7.15073809719169853341426010700, 7.85514777174456631532685033025, 8.878072131534690514249358187571, 9.482632250871599528466865055640