L(s) = 1 | + (1 + i)2-s + (0.561 − 0.561i)3-s + 2i·4-s + (−4.87 − 1.10i)5-s + 1.12·6-s + (−8.38 − 8.38i)7-s + (−2 + 2i)8-s + 8.36i·9-s + (−3.77 − 5.97i)10-s − 11.2·11-s + (1.12 + 1.12i)12-s + (−15.9 + 15.9i)13-s − 16.7i·14-s + (−3.35 + 2.12i)15-s − 4·16-s + (14.2 + 14.2i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (0.187 − 0.187i)3-s + 0.5i·4-s + (−0.975 − 0.220i)5-s + 0.187·6-s + (−1.19 − 1.19i)7-s + (−0.250 + 0.250i)8-s + 0.929i·9-s + (−0.377 − 0.597i)10-s − 1.02·11-s + (0.0936 + 0.0936i)12-s + (−1.22 + 1.22i)13-s − 1.19i·14-s + (−0.223 + 0.141i)15-s − 0.250·16-s + (0.840 + 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0148753 - 0.0889007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0148753 - 0.0889007i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - i)T \) |
| 5 | \( 1 + (4.87 + 1.10i)T \) |
| 23 | \( 1 + (3.39 - 3.39i)T \) |
good | 3 | \( 1 + (-0.561 + 0.561i)T - 9iT^{2} \) |
| 7 | \( 1 + (8.38 + 8.38i)T + 49iT^{2} \) |
| 11 | \( 1 + 11.2T + 121T^{2} \) |
| 13 | \( 1 + (15.9 - 15.9i)T - 169iT^{2} \) |
| 17 | \( 1 + (-14.2 - 14.2i)T + 289iT^{2} \) |
| 19 | \( 1 + 35.4iT - 361T^{2} \) |
| 29 | \( 1 + 26.2iT - 841T^{2} \) |
| 31 | \( 1 - 19.1T + 961T^{2} \) |
| 37 | \( 1 + (26.7 + 26.7i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 19.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (44.0 - 44.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (36.1 + 36.1i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (0.502 - 0.502i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 5.81iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 0.344T + 3.72e3T^{2} \) |
| 67 | \( 1 + (17.5 + 17.5i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 14.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (68.0 - 68.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 82.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (32.8 - 32.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 59.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-33.5 - 33.5i)T + 9.40e3iT^{2} \) |
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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
−12.83745647025577681488101531181, −11.71684263469988121918964672029, −10.63563324630596982367373468448, −9.610964827052234128488353087276, −8.188519958403216291401160126071, −7.37145503520473435564835877649, −6.77689474009757637325577532055, −5.01991900610686310724264124227, −4.13599923163581377512106852346, −2.79993247377135774267530920402,
0.03775550455115702524712451396, 2.93014369341903769126145044156, 3.32950847649151622376215845867, 5.07743375286579831775219726416, 6.06094193287075217419082509231, 7.43536412200146984708487513847, 8.551904600446263072977418404470, 9.870321015291401879748350748111, 10.26152477450190125952845385724, 11.99488889679371719923127745046