L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.58·5-s + (−2.64 − 0.0963i)7-s + 0.999·8-s + (0.794 − 1.37i)10-s + 1.58·11-s + (2.40 − 4.16i)13-s + (1.40 − 2.24i)14-s + (−0.5 + 0.866i)16-s + (2.69 − 4.67i)17-s + (−3.54 − 6.14i)19-s + (0.794 + 1.37i)20-s + (−0.794 + 1.37i)22-s − 0.300·23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.710·5-s + (−0.999 − 0.0364i)7-s + 0.353·8-s + (0.251 − 0.434i)10-s + 0.478·11-s + (0.667 − 1.15i)13-s + (0.375 − 0.599i)14-s + (−0.125 + 0.216i)16-s + (0.654 − 1.13i)17-s + (−0.814 − 1.41i)19-s + (0.177 + 0.307i)20-s + (−0.169 + 0.293i)22-s − 0.0626·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.519194 - 0.346064i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.519194 - 0.346064i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.0963i)T \) |
good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.69 + 4.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.300T + 23T^{2} \) |
| 29 | \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.35 - 2.34i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.93 - 5.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.23 - 5.60i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.23 + 3.87i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.02 - 8.70i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + (-8.02 + 13.9i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.19 - 7.26i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.18 + 2.04i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.60 + 2.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.14733736667289451866259235613, −10.08938640986445483370144474859, −9.297056691059508565107549911203, −8.338357570662908724774989279574, −7.42667138031756938272507302555, −6.55167411460536584485989717777, −5.55940084735348107065927736712, −4.19174952634560981867959741388, −2.99197892063502348043752662188, −0.47917294819184509109412475082,
1.72543276160858166446234379667, 3.60050624706311344299279063644, 3.99317401508695538031094859079, 5.89381929066827786350808604127, 6.84660254686393453745340411892, 8.040453910574620024323675229422, 8.847915687339621451727062839948, 9.769661439535378098621529712381, 10.61904579019137042977631485427, 11.54317444974217256483310657108