L(s) = 1 | − 2.10·2-s − 3-s + 2.42·4-s + 5-s + 2.10·6-s − 4.09·7-s − 0.894·8-s + 9-s − 2.10·10-s + 1.84·11-s − 2.42·12-s + 13-s + 8.62·14-s − 15-s − 2.96·16-s − 7.44·17-s − 2.10·18-s + 6.10·19-s + 2.42·20-s + 4.09·21-s − 3.89·22-s + 3.27·23-s + 0.894·24-s + 25-s − 2.10·26-s − 27-s − 9.94·28-s + ⋯ |
L(s) = 1 | − 1.48·2-s − 0.577·3-s + 1.21·4-s + 0.447·5-s + 0.858·6-s − 1.54·7-s − 0.316·8-s + 0.333·9-s − 0.665·10-s + 0.557·11-s − 0.700·12-s + 0.277·13-s + 2.30·14-s − 0.258·15-s − 0.742·16-s − 1.80·17-s − 0.495·18-s + 1.40·19-s + 0.542·20-s + 0.894·21-s − 0.829·22-s + 0.683·23-s + 0.182·24-s + 0.200·25-s − 0.412·26-s − 0.192·27-s − 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 7 | \( 1 + 4.09T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 17 | \( 1 + 7.44T + 17T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 + 1.78T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.27T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 3.40T + 71T^{2} \) |
| 73 | \( 1 + 5.41T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 7.99T + 97T^{2} \) |
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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
−7.56330012126249698142543015476, −7.13418704240960905796709782984, −6.42768575042838063036672019526, −6.00941968633983384461154621806, −4.94496186400750795760322500184, −3.95489512799172429945276416827, −2.98669846569732302312608997262, −1.99608698102636699570382925203, −0.966337714548354592277613980270, 0,
0.966337714548354592277613980270, 1.99608698102636699570382925203, 2.98669846569732302312608997262, 3.95489512799172429945276416827, 4.94496186400750795760322500184, 6.00941968633983384461154621806, 6.42768575042838063036672019526, 7.13418704240960905796709782984, 7.56330012126249698142543015476