| L(s) = 1 | + (2.50 − 0.571i)2-s + (0.445 + 1.44i)3-s + (2.33 − 1.12i)4-s + (−3.49 − 1.37i)5-s + (1.93 + 3.35i)6-s + (−3.35 − 1.93i)7-s + (−2.82 + 2.25i)8-s + (5.55 − 3.78i)9-s + (−9.53 − 1.43i)10-s + (−2.36 − 1.14i)11-s + (2.66 + 2.86i)12-s + (4.94 − 0.745i)13-s + (−9.50 − 2.93i)14-s + (0.424 − 5.66i)15-s + (−12.2 + 15.3i)16-s + (8.80 + 22.4i)17-s + ⋯ |
| L(s) = 1 | + (1.25 − 0.285i)2-s + (0.148 + 0.481i)3-s + (0.583 − 0.281i)4-s + (−0.699 − 0.274i)5-s + (0.323 + 0.559i)6-s + (−0.479 − 0.276i)7-s + (−0.353 + 0.281i)8-s + (0.616 − 0.420i)9-s + (−0.953 − 0.143i)10-s + (−0.215 − 0.103i)11-s + (0.221 + 0.239i)12-s + (0.380 − 0.0573i)13-s + (−0.679 − 0.209i)14-s + (0.0282 − 0.377i)15-s + (−0.765 + 0.960i)16-s + (0.517 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.71480 - 0.0637238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71480 - 0.0637238i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (41.1 + 12.6i)T \) |
| good | 2 | \( 1 + (-2.50 + 0.571i)T + (3.60 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.445 - 1.44i)T + (-7.43 + 5.06i)T^{2} \) |
| 5 | \( 1 + (3.49 + 1.37i)T + (18.3 + 17.0i)T^{2} \) |
| 7 | \( 1 + (3.35 + 1.93i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (2.36 + 1.14i)T + (75.4 + 94.6i)T^{2} \) |
| 13 | \( 1 + (-4.94 + 0.745i)T + (161. - 49.8i)T^{2} \) |
| 17 | \( 1 + (-8.80 - 22.4i)T + (-211. + 196. i)T^{2} \) |
| 19 | \( 1 + (-3.29 + 4.83i)T + (-131. - 336. i)T^{2} \) |
| 23 | \( 1 + (0.176 + 2.35i)T + (-523. + 78.8i)T^{2} \) |
| 29 | \( 1 + (-9.16 + 29.7i)T + (-694. - 473. i)T^{2} \) |
| 31 | \( 1 + (4.20 - 3.89i)T + (71.8 - 958. i)T^{2} \) |
| 37 | \( 1 + (-59.7 + 34.4i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-7.95 - 34.8i)T + (-1.51e3 + 729. i)T^{2} \) |
| 47 | \( 1 + (68.9 - 33.2i)T + (1.37e3 - 1.72e3i)T^{2} \) |
| 53 | \( 1 + (65.3 + 9.85i)T + (2.68e3 + 827. i)T^{2} \) |
| 59 | \( 1 + (60.3 - 75.7i)T + (-774. - 3.39e3i)T^{2} \) |
| 61 | \( 1 + (-42.8 + 46.2i)T + (-278. - 3.71e3i)T^{2} \) |
| 67 | \( 1 + (-101. - 69.2i)T + (1.64e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (28.8 + 2.16i)T + (4.98e3 + 751. i)T^{2} \) |
| 73 | \( 1 + (1.83 + 12.1i)T + (-5.09e3 + 1.57e3i)T^{2} \) |
| 79 | \( 1 + (2.10 - 3.63i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-43.1 + 13.3i)T + (5.69e3 - 3.88e3i)T^{2} \) |
| 89 | \( 1 + (-28.8 - 93.5i)T + (-6.54e3 + 4.46e3i)T^{2} \) |
| 97 | \( 1 + (-16.5 - 7.96i)T + (5.86e3 + 7.35e3i)T^{2} \) |
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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
−15.45147908843368717424829616285, −14.59435677351399903859724166463, −13.20298149020851926579087660039, −12.50037191438197319466688135963, −11.26399124458879303885577541023, −9.807751210780688110446735805481, −8.186009463458691542036325591744, −6.21712282316291985404103224253, −4.45248286675681487811632721212, −3.50031770876672061637377898146,
3.26357156522789707404691387043, 4.92380598620503342068540138386, 6.56986381317542797360108165412, 7.73244096450007893103629013655, 9.659505029759550863500439370753, 11.49164457979783366445023936183, 12.56963644724530058400795557991, 13.42273582223027846658203199867, 14.44358412762223922234827069133, 15.62842770631954924905555353515
