L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.448 + 1.67i)3-s + (0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s + (−0.448 − 1.67i)6-s + (−0.5 + 0.866i)7-s + 0.999i·8-s + (−1.73 − 1.00i)9-s + (−0.707 + 0.707i)10-s + (1.22 + 1.22i)12-s + (−0.965 + 0.258i)13-s − 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (−1.93 + 0.517i)19-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.448 + 1.67i)3-s + (0.499 − 0.866i)4-s + (0.965 − 0.258i)5-s + (−0.448 − 1.67i)6-s + (−0.5 + 0.866i)7-s + 0.999i·8-s + (−1.73 − 1.00i)9-s + (−0.707 + 0.707i)10-s + (1.22 + 1.22i)12-s + (−0.965 + 0.258i)13-s − 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (−1.93 + 0.517i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2263521407\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2263521407\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.965 - 0.258i)T \) |
good | 3 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (1.93 - 0.517i)T + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.308750923229462404583520118588, −8.835482149003047374043948732137, −8.173302790330174545703563002319, −6.80543664362327918475350548036, −6.12095979846511608856864032641, −5.64794641134807887109069370603, −4.92783358785647678229052317823, −4.18783531221322329140912034784, −2.78601293417513393599404779054, −2.00180241635514671503896809508,
0.17223536987829894474765051419, 1.39688249620217045515559152296, 2.24094390839627921025791651298, 2.81921363172688754402150967009, 4.17907461328402439334432187448, 5.47777269654889209641637989939, 6.48482889667700961803693044674, 6.69560320858126997835198142122, 7.36906568565406614933955331011, 8.077916299543545551617164913098