L(s) = 1 | + (−1.66 − 1.16i)2-s + (−1.43 + 0.967i)3-s + (0.729 + 2.00i)4-s + (−1.48 − 1.67i)5-s + (3.52 + 0.0638i)6-s + (1.13 + 2.43i)7-s + (0.0700 − 0.261i)8-s + (1.12 − 2.78i)9-s + (0.518 + 4.51i)10-s + (4.08 + 4.86i)11-s + (−2.98 − 2.17i)12-s + (1.19 + 1.70i)13-s + (0.948 − 5.37i)14-s + (3.74 + 0.968i)15-s + (2.84 − 2.38i)16-s + (−0.174 − 0.649i)17-s + ⋯ |
L(s) = 1 | + (−1.17 − 0.824i)2-s + (−0.829 + 0.558i)3-s + (0.364 + 1.00i)4-s + (−0.663 − 0.748i)5-s + (1.43 + 0.0260i)6-s + (0.429 + 0.920i)7-s + (0.0247 − 0.0923i)8-s + (0.375 − 0.926i)9-s + (0.164 + 1.42i)10-s + (1.23 + 1.46i)11-s + (−0.862 − 0.627i)12-s + (0.331 + 0.473i)13-s + (0.253 − 1.43i)14-s + (0.968 + 0.250i)15-s + (0.711 − 0.597i)16-s + (−0.0422 − 0.157i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.476i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.398734 + 0.101220i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.398734 + 0.101220i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.43 - 0.967i)T \) |
| 5 | \( 1 + (1.48 + 1.67i)T \) |
good | 2 | \( 1 + (1.66 + 1.16i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-1.13 - 2.43i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-4.08 - 4.86i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.19 - 1.70i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.174 + 0.649i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (2.70 - 1.56i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.409 + 0.191i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.39 - 7.90i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.53 - 0.559i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.76 + 0.739i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (9.28 + 1.63i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.307 + 3.51i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-5.68 + 2.65i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-5.99 - 5.99i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.07 - 5.10i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.53 + 0.560i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-4.38 + 3.06i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-2.05 - 1.18i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.33 + 0.358i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.73 - 1.18i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.27 - 3.24i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-0.428 - 0.742i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.88 + 0.340i)T + (95.5 + 16.8i)T^{2} \) |
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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
−12.39981175902006334103356277215, −12.06567625592168739470294742897, −11.32088112064610098878383798387, −10.19967853665069325680218195107, −9.140582051689043091541889028555, −8.675637036886320688407125831426, −7.03035132084847953537478623001, −5.30119128651141422038109863753, −4.07511844889789726858841428640, −1.60159880379706335200393764237,
0.76093154720818197223465605621, 3.93161198567752449342384019671, 6.09174238182417213184798098289, 6.76498257371910312166265773301, 7.78098278329640165097536552526, 8.522100048475528898343458267598, 10.16911790539962037675457072496, 11.02540549789795397845823799874, 11.69445370072614529001186858505, 13.25746107350290145300189072954