L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 0.999·10-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.809 + 0.587i)18-s + (−1.61 + 1.17i)19-s − 23-s + (0.309 − 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)5-s + (−0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s − 0.999·10-s + (−0.809 − 0.587i)15-s + (−0.309 − 0.951i)16-s + (−0.309 − 0.951i)17-s + (0.809 + 0.587i)18-s + (−1.61 + 1.17i)19-s − 23-s + (0.309 − 0.951i)24-s + (−0.809 + 0.587i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5181395654\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5181395654\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−9.944444404088392313046617006436, −9.171806995647577889552302888288, −8.299055088599953244577528281942, −7.37801301636626056445797853938, −6.68947246704710108807794324813, −6.10754269927697927744275290028, −5.51229407701784984901027060648, −4.34605032716955185873254304080, −3.33603143534290761653263661612, −2.16787952841388785942774609021,
0.43804744313907879574743456797, 1.77382117077622786163258292701, 2.30614400051045991385911962530, 3.97883713395728400371461000770, 4.77606064599277148079940644486, 5.99488215047517627284129987005, 6.23873478839760671829413999434, 7.35133289214317143407147929714, 8.431336828202679413697860193370, 8.976763798636364562954999534639