L(s) = 1 | + (2.07 + 1.19i)2-s + (0.641 − 1.60i)3-s + (1.85 + 3.22i)4-s + (−0.854 − 0.229i)5-s + (3.25 − 2.56i)6-s + (−2.38 + 0.639i)7-s + 4.11i·8-s + (−2.17 − 2.06i)9-s + (−1.49 − 1.49i)10-s + (−2.06 + 0.553i)11-s + (6.37 − 0.927i)12-s + (0.147 + 0.254i)13-s + (−5.70 − 1.52i)14-s + (−0.916 + 1.22i)15-s + (−1.19 + 2.07i)16-s + (3.94 + 1.20i)17-s + ⋯ |
L(s) = 1 | + (1.46 + 0.845i)2-s + (0.370 − 0.928i)3-s + (0.929 + 1.61i)4-s + (−0.382 − 0.102i)5-s + (1.32 − 1.04i)6-s + (−0.901 + 0.241i)7-s + 1.45i·8-s + (−0.726 − 0.687i)9-s + (−0.473 − 0.473i)10-s + (−0.622 + 0.166i)11-s + (1.84 − 0.267i)12-s + (0.0408 + 0.0707i)13-s + (−1.52 − 0.408i)14-s + (−0.236 + 0.317i)15-s + (−0.299 + 0.518i)16-s + (0.956 + 0.291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.13130 + 0.523251i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.13130 + 0.523251i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.641 + 1.60i)T \) |
| 17 | \( 1 + (-3.94 - 1.20i)T \) |
good | 2 | \( 1 + (-2.07 - 1.19i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.854 + 0.229i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (2.38 - 0.639i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (2.06 - 0.553i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.147 - 0.254i)T + (-6.5 + 11.2i)T^{2} \) |
| 19 | \( 1 - 6.94iT - 19T^{2} \) |
| 23 | \( 1 + (-1.67 + 6.25i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (0.853 + 3.18i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-3.61 - 0.968i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (1.49 - 1.49i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.90 + 7.11i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-9.64 - 5.56i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.03 - 5.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 + (5.41 - 3.12i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.22 + 2.47i)T + (52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (0.110 + 0.191i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.53 - 7.53i)T - 71iT^{2} \) |
| 73 | \( 1 + (9.37 - 9.37i)T - 73iT^{2} \) |
| 79 | \( 1 + (10.8 - 2.90i)T + (68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-2.97 - 1.71i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.17T + 89T^{2} \) |
| 97 | \( 1 + (-2.36 - 8.83i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−12.93499222444491827965374965048, −12.58940413253530779604546596621, −11.80925483503126687017758849999, −9.989398587320911086382668317855, −8.294053129439436941993826018806, −7.55477635150603785580452919513, −6.40066842275525374505666415201, −5.68858349184894621592229983968, −3.99292823583181451115149491053, −2.81370635501098067472873597245,
2.85586014435445456141530289522, 3.57462377452206606443476979870, 4.81232495722088252747328918068, 5.80931386084823509340126275450, 7.49111775124721895968914642109, 9.208173597611062599802374613582, 10.19977198123660905358732020550, 11.06611654316039695951957661052, 11.88002047000784506072538907221, 13.18040285425983878256805168289