L(s) = 1 | + (−0.142 − 0.989i)3-s + (3.91 + 1.15i)5-s + (−0.0825 + 0.0952i)7-s + (−0.959 + 0.281i)9-s + (−1.03 + 0.667i)11-s + (1.25 + 1.44i)13-s + (0.581 − 4.04i)15-s + (0.787 − 1.72i)17-s + (0.0613 + 0.134i)19-s + (0.106 + 0.0681i)21-s + (−2.15 − 4.28i)23-s + (9.83 + 6.31i)25-s + (0.415 + 0.909i)27-s + (3.11 − 6.82i)29-s + (0.335 − 2.33i)31-s + ⋯ |
L(s) = 1 | + (−0.0821 − 0.571i)3-s + (1.75 + 0.514i)5-s + (−0.0311 + 0.0359i)7-s + (−0.319 + 0.0939i)9-s + (−0.313 + 0.201i)11-s + (0.347 + 0.400i)13-s + (0.150 − 1.04i)15-s + (0.190 − 0.418i)17-s + (0.0140 + 0.0308i)19-s + (0.0231 + 0.0148i)21-s + (−0.448 − 0.893i)23-s + (1.96 + 1.26i)25-s + (0.0799 + 0.175i)27-s + (0.578 − 1.26i)29-s + (0.0602 − 0.419i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56371 - 0.163430i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56371 - 0.163430i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (2.15 + 4.28i)T \) |
good | 5 | \( 1 + (-3.91 - 1.15i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.0825 - 0.0952i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (1.03 - 0.667i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 1.44i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.787 + 1.72i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.0613 - 0.134i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.11 + 6.82i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.335 + 2.33i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (7.41 - 2.17i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (10.0 + 2.95i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.71 - 11.9i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 5.71T + 47T^{2} \) |
| 53 | \( 1 + (4.37 - 5.04i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (4.98 + 5.75i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (0.180 - 1.25i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (8.94 + 5.75i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-7.77 - 4.99i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.25 + 11.4i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-2.44 - 2.81i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-7.82 + 2.29i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.00 - 6.98i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-12.7 - 3.73i)T + (81.6 + 52.4i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93217828289917470576123054378, −10.76676055260101462262436650188, −9.985751294765142475689551173595, −9.156526032253820622406916633320, −7.916808474846965991987680048422, −6.59263382916107703156267637273, −6.12044546319307270417092378178, −4.90637258544649506170672908496, −2.85476536813044432320473613097, −1.75560068291132735179469174889,
1.71578711285566175275453896510, 3.34091520610292365253038656498, 5.07497521153060764648621000720, 5.62927267089568986221810171467, 6.73330999435376705734185587274, 8.417088909683587155941549421055, 9.130832280402774491839574698619, 10.19159538675990634112124721356, 10.51663852420488196382265258659, 11.97562144064522533571982812044