L(s) = 1 | + 2·2-s + (1.23 + 0.715i)3-s + 4·4-s + (−3.73 + 6.47i)5-s + (2.47 + 1.43i)6-s − 3.76i·7-s + 8·8-s + (−3.47 − 6.02i)9-s + (−7.47 + 12.9i)10-s − 14.0i·11-s + (4.95 + 2.86i)12-s + (−3 − 5.19i)13-s − 7.53i·14-s + (−9.26 + 5.34i)15-s + 16·16-s + (−13.4 + 23.3i)17-s + ⋯ |
L(s) = 1 | + 2-s + (0.412 + 0.238i)3-s + 4-s + (−0.747 + 1.29i)5-s + (0.412 + 0.238i)6-s − 0.537i·7-s + 8-s + (−0.386 − 0.669i)9-s + (−0.747 + 1.29i)10-s − 1.27i·11-s + (0.412 + 0.238i)12-s + (−0.230 − 0.399i)13-s − 0.537i·14-s + (−0.617 + 0.356i)15-s + 16-s + (−0.792 + 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.12180 + 0.414691i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12180 + 0.414691i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 19 | \( 1 + (-18.7 + 3.27i)T \) |
good | 3 | \( 1 + (-1.23 - 0.715i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (3.73 - 6.47i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 3.76iT - 49T^{2} \) |
| 11 | \( 1 + 14.0iT - 121T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (13.4 - 23.3i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (21.6 - 12.4i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-1.73 - 3.01i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 39.4iT - 961T^{2} \) |
| 37 | \( 1 - 3.38T + 1.36e3T^{2} \) |
| 41 | \( 1 + (4.45 - 7.71i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (45.3 + 26.2i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-19.1 + 11.0i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-41.9 - 72.6i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (22.2 + 12.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (13.7 + 23.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-32.0 + 18.4i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-89.3 - 51.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-4.45 + 7.71i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (111. + 64.6i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 27.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (64.2 + 111. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (53.0 - 91.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.29194265867851138589051244953, −13.63079232813123628297919292901, −12.07675712679290601948273326017, −11.13474355151864456452129002761, −10.33914208074678793775420571991, −8.319770056994913734956758084039, −7.08132331610636919852417243452, −5.95838159015304368150552002255, −3.83287573508576673010953296209, −3.11555682435495311365076196549,
2.28454488513586563838736527553, 4.37251911802085148314727859887, 5.26597033623146415118511824267, 7.20492768639750518730099848790, 8.220506355573729991745796987706, 9.599055422352761635421039112881, 11.53083895679186798893272044775, 12.12707734898910135042538915332, 13.10494291172818084771401800337, 14.03885158090236371272430362314