L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.681 + 1.59i)3-s + (0.866 + 0.499i)4-s + (1.33 + 1.79i)5-s + (1.07 − 1.36i)6-s + (1.22 + 4.57i)7-s + (−0.707 − 0.707i)8-s + (−2.07 − 2.16i)9-s + (−0.821 − 2.07i)10-s + (0.957 + 0.256i)11-s + (−1.38 + 1.03i)12-s + (3.59 − 0.326i)13-s − 4.73i·14-s + (−3.76 + 0.898i)15-s + (0.500 + 0.866i)16-s + (−6.31 − 3.64i)17-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (−0.393 + 0.919i)3-s + (0.433 + 0.249i)4-s + (0.595 + 0.803i)5-s + (0.436 − 0.556i)6-s + (0.463 + 1.72i)7-s + (−0.249 − 0.249i)8-s + (−0.690 − 0.723i)9-s + (−0.259 − 0.657i)10-s + (0.288 + 0.0773i)11-s + (−0.400 + 0.299i)12-s + (0.995 − 0.0906i)13-s − 1.26i·14-s + (−0.972 + 0.231i)15-s + (0.125 + 0.216i)16-s + (−1.53 − 0.884i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.456539 + 0.825904i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.456539 + 0.825904i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.681 - 1.59i)T \) |
| 5 | \( 1 + (-1.33 - 1.79i)T \) |
| 13 | \( 1 + (-3.59 + 0.326i)T \) |
good | 7 | \( 1 + (-1.22 - 4.57i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.957 - 0.256i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (6.31 + 3.64i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.503 - 1.88i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.14 + 1.81i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.96 - 1.70i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.89 + 2.89i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.31 + 1.15i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.42 - 5.33i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (3.67 - 6.37i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.710 + 0.710i)T + 47iT^{2} \) |
| 53 | \( 1 - 9.89T + 53T^{2} \) |
| 59 | \( 1 + (0.358 + 1.33i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.47 + 6.02i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.67 + 10.0i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.344 - 0.0923i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-8.41 + 8.41i)T - 73iT^{2} \) |
| 79 | \( 1 - 0.626T + 79T^{2} \) |
| 83 | \( 1 + (-2.20 + 2.20i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.91 - 0.513i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (10.6 - 2.86i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.29648854101890110180531901748, −10.85444728115713559960250338178, −9.655323966590673779076869023770, −9.090446531739165461924411584836, −8.362656149632517939913497593012, −6.67731102263723830846352156289, −5.97083481100886438179901362753, −4.92556408275378892466621542890, −3.25611067674030808425475969264, −2.15491871612766527291996393163,
0.855017422375750408545534602494, 1.82502733365186849216861048720, 4.09471140249293936849758656912, 5.37488234297855133418991485191, 6.57557042795221740038984038792, 7.12697901876250504729650021072, 8.334235429713307422156108533382, 8.851429947389130014022555314678, 10.27911565375841980289795021471, 10.91514393054906417649427938810