L(s) = 1 | + 2·5-s + 2·7-s − 6·11-s − 2·13-s − 23-s − 25-s − 6·29-s − 8·31-s + 4·35-s − 10·41-s + 12·43-s − 8·47-s − 3·49-s − 2·53-s − 12·55-s − 12·59-s + 4·61-s − 4·65-s + 12·67-s − 10·73-s − 12·77-s + 6·79-s + 14·83-s − 4·91-s − 6·97-s + 6·101-s − 14·103-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.755·7-s − 1.80·11-s − 0.554·13-s − 0.208·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.676·35-s − 1.56·41-s + 1.82·43-s − 1.16·47-s − 3/7·49-s − 0.274·53-s − 1.61·55-s − 1.56·59-s + 0.512·61-s − 0.496·65-s + 1.46·67-s − 1.17·73-s − 1.36·77-s + 0.675·79-s + 1.53·83-s − 0.419·91-s − 0.609·97-s + 0.597·101-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.009088544194373664704576673472, −7.74940445665935159391608281760, −6.81404055785242482314012546704, −5.70446099983308366623422114205, −5.34329120767643148260995938996, −4.61781233004508478544485198077, −3.38807414963777119141033594454, −2.33137889789155181282145246786, −1.75673035985210661335841948053, 0,
1.75673035985210661335841948053, 2.33137889789155181282145246786, 3.38807414963777119141033594454, 4.61781233004508478544485198077, 5.34329120767643148260995938996, 5.70446099983308366623422114205, 6.81404055785242482314012546704, 7.74940445665935159391608281760, 8.009088544194373664704576673472