L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (2 − i)5-s + (2 + 2i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 2.12i)10-s − 4.82i·11-s + (−1.41 + 1.41i)13-s + 2.82·14-s − 1.00·16-s + (2.58 − 2.58i)17-s + 0.585i·19-s + (−1.00 − 2.00i)20-s + (−3.41 − 3.41i)22-s + (−0.707 − 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.894 − 0.447i)5-s + (0.755 + 0.755i)7-s + (−0.250 − 0.250i)8-s + (0.223 − 0.670i)10-s − 1.45i·11-s + (−0.392 + 0.392i)13-s + 0.755·14-s − 0.250·16-s + (0.627 − 0.627i)17-s + 0.134i·19-s + (−0.223 − 0.447i)20-s + (−0.727 − 0.727i)22-s + (−0.147 − 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.930718158\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.930718158\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 13 | \( 1 + (1.41 - 1.41i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.58 + 2.58i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.585iT - 19T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 + (1.41 + 1.41i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.41iT - 41T^{2} \) |
| 43 | \( 1 + (-2.24 + 2.24i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-8.65 - 8.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 5.41T + 61T^{2} \) |
| 67 | \( 1 + (-6.82 - 6.82i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.41iT - 71T^{2} \) |
| 73 | \( 1 + (9.82 - 9.82i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.17iT - 79T^{2} \) |
| 83 | \( 1 + (0.757 + 0.757i)T + 83iT^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + (9.07 + 9.07i)T + 97iT^{2} \) |
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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
−8.882388103273202617267121884521, −8.505477100494603611561695986360, −7.34952851975266772335905539062, −6.20445853765990796355416902242, −5.54501149760355925354190724719, −5.11189537985212356568451467188, −4.01574966591202223595438963485, −2.85251397147586297028497444578, −2.07173844550366394264430524610, −0.930310970569492929589868957302,
1.50090648920131694298948821588, 2.47589051686666017748909625593, 3.67041853933436641896091249608, 4.63944127215492929710066738627, 5.24175421648671741620662862508, 6.16871962481573522203010619796, 7.00666759987097032629595362727, 7.54594837557895853331447400969, 8.276768773591905981790519605790, 9.437607375743435729235553802006