L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + (0.392 − 2.20i)5-s + (0.499 − 0.866i)6-s + (−3.01 − 3.01i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.948 + 2.02i)10-s + 0.889·11-s + (−0.707 + 0.707i)12-s + (−3.48 + 0.935i)13-s + (2.12 + 3.68i)14-s + (2.02 + 0.948i)15-s + (0.500 + 0.866i)16-s + (−2.12 + 7.93i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + (0.175 − 0.984i)5-s + (0.204 − 0.353i)6-s + (−1.13 − 1.13i)7-s + (−0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.299 + 0.640i)10-s + 0.268·11-s + (−0.204 + 0.204i)12-s + (−0.967 + 0.259i)13-s + (0.569 + 0.985i)14-s + (0.522 + 0.244i)15-s + (0.125 + 0.216i)16-s + (−0.515 + 1.92i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0174801 + 0.0673066i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0174801 + 0.0673066i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (-0.392 + 2.20i)T \) |
| 19 | \( 1 + (1.13 - 4.20i)T \) |
good | 7 | \( 1 + (3.01 + 3.01i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.889T + 11T^{2} \) |
| 13 | \( 1 + (3.48 - 0.935i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (2.12 - 7.93i)T + (-14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (-0.772 - 2.88i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.84 - 6.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.64iT - 31T^{2} \) |
| 37 | \( 1 + (-3.39 + 3.39i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.93 + 2.84i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (11.1 + 2.98i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (9.22 - 2.47i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-10.6 + 2.86i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.41 + 2.45i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.57 - 4.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.57 + 5.87i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.67 - 0.965i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.11 + 0.835i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.06 + 8.77i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.14 - 4.14i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.50 - 7.80i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.59 - 0.426i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.76481804928314478202347147826, −10.08770684422813896711113413275, −9.523551661646375512413247785666, −8.694925682459896192967076248568, −7.69254267158039632202355836494, −6.64329118891295265009941982396, −5.69698688319846534058868895288, −4.26572841382502646316425969239, −3.58579162421346570683642345673, −1.68510758953209204416622625517,
0.04701410789876875616374893194, 2.46570307989251521312232035906, 2.88037001715341623329970244776, 5.04878264981230202454982022883, 6.20010345033298446170488073827, 6.78565243854801312547442560222, 7.46647051030370974992235712168, 8.745943902590476086411644723485, 9.541902579946373152724977532881, 10.08712393326027373518824899878