L(s) = 1 | + (0.258 + 0.965i)3-s + (0.965 − 0.258i)5-s + (−0.5 + 0.866i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.448i)11-s + (0.499 + 0.866i)15-s + (−0.965 − 0.258i)21-s + (0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s + 1.93i·29-s + (0.866 + 0.5i)31-s + (−0.5 − 0.133i)33-s + (−0.258 + 0.965i)35-s + (−0.707 + 0.707i)45-s + (−0.499 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)3-s + (0.965 − 0.258i)5-s + (−0.5 + 0.866i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.448i)11-s + (0.499 + 0.866i)15-s + (−0.965 − 0.258i)21-s + (0.866 − 0.499i)25-s + (−0.707 − 0.707i)27-s + 1.93i·29-s + (0.866 + 0.5i)31-s + (−0.5 − 0.133i)33-s + (−0.258 + 0.965i)35-s + (−0.707 + 0.707i)45-s + (−0.499 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.385835012\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.385835012\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 11 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 1.93iT - T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
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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
−9.162765311716085343480964611861, −8.621184667790695377625534291779, −7.68516582184442202401691378711, −6.54674661540946788892099844690, −5.93860068947270566991855730831, −5.08517995057418527821256362414, −4.68817324806403678348751852929, −3.34397243797260667321031606728, −2.73745233261145963043178251221, −1.73576112211095175568417285951,
0.77996086352470450185359195286, 1.98148316744802828220662861454, 2.81196752201371823567877596900, 3.64120451560086000568848870131, 4.81420019699250264881891877449, 5.93464384580302637335677179502, 6.31706953199102309159836255230, 6.99299821676925465431646606157, 7.85579649384094036986209697756, 8.343078555335708236419693811906