L(s) = 1 | + (0.965 + 0.258i)2-s + i·3-s + (0.866 + 0.499i)4-s + (0.421 − 0.112i)5-s + (−0.258 + 0.965i)6-s + (2.44 + 1.01i)7-s + (0.707 + 0.707i)8-s − 9-s + 0.435·10-s + (1.13 + 1.13i)11-s + (−0.499 + 0.866i)12-s + (−1.06 − 3.44i)13-s + (2.09 + 1.61i)14-s + (0.112 + 0.421i)15-s + (0.500 + 0.866i)16-s + (0.582 − 1.00i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + 0.577i·3-s + (0.433 + 0.249i)4-s + (0.188 − 0.0504i)5-s + (−0.105 + 0.394i)6-s + (0.923 + 0.382i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s + 0.137·10-s + (0.342 + 0.342i)11-s + (−0.144 + 0.249i)12-s + (−0.294 − 0.955i)13-s + (0.561 + 0.430i)14-s + (0.0291 + 0.108i)15-s + (0.125 + 0.216i)16-s + (0.141 − 0.244i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07816 + 1.20414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07816 + 1.20414i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-2.44 - 1.01i)T \) |
| 13 | \( 1 + (1.06 + 3.44i)T \) |
good | 5 | \( 1 + (-0.421 + 0.112i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.13 - 1.13i)T + 11iT^{2} \) |
| 17 | \( 1 + (-0.582 + 1.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.57 - 4.57i)T + 19iT^{2} \) |
| 23 | \( 1 + (4.85 - 2.80i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.92 - 3.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.28 + 8.51i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.27 + 4.75i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.78 + 0.478i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (0.251 - 0.145i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.19 - 4.45i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.94 + 5.10i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.35 + 12.5i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 0.413iT - 61T^{2} \) |
| 67 | \( 1 + (1.22 - 1.22i)T - 67iT^{2} \) |
| 71 | \( 1 + (7.98 + 2.13i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.797 - 0.213i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.79 + 8.30i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.03 + 7.03i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.66 - 1.78i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.433 + 1.61i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.14093433013424887204935835360, −10.02898914530384930222712458697, −9.361401360751074720836770977749, −7.994792161870790042923975808084, −7.56675404668787516065781750895, −5.89694607394990459556172057317, −5.44365386249805867127789308131, −4.37538520849658770849994626672, −3.33716928961606332642437662324, −1.89920729236626309922963002973,
1.35789824949306518013654124536, 2.55687199109741235795247897658, 4.02183346117751847937539681853, 4.94521487689308050671381599736, 6.05348143721031117987126153669, 6.93880079865209409691114658961, 7.80047396609241968897981892961, 8.805521954254993574839963290248, 9.931320940827915293938312815155, 10.90490189844393929658667135820