| L(s) = 1 | − 3.04·5-s − 7-s + 4.77·11-s − 5.27·13-s − 0.418·17-s + 6.54·19-s + 8.71·23-s + 4.27·25-s + 3.88·29-s − 6·31-s + 3.04·35-s − 11.5·37-s − 6.92·41-s − 6.27·43-s + 3.46·47-s + 49-s − 10.8·53-s − 14.5·55-s − 8.71·59-s − 11.2·61-s + 16.0·65-s + 0.274·67-s + 8.24·71-s − 2.72·73-s − 4.77·77-s − 3.72·79-s + 0.837·83-s + ⋯ |
| L(s) = 1 | − 1.36·5-s − 0.377·7-s + 1.44·11-s − 1.46·13-s − 0.101·17-s + 1.50·19-s + 1.81·23-s + 0.854·25-s + 0.721·29-s − 1.07·31-s + 0.514·35-s − 1.89·37-s − 1.08·41-s − 0.956·43-s + 0.505·47-s + 0.142·49-s − 1.49·53-s − 1.96·55-s − 1.13·59-s − 1.44·61-s + 1.99·65-s + 0.0335·67-s + 0.978·71-s − 0.318·73-s − 0.544·77-s − 0.419·79-s + 0.0919·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.04T + 5T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 + 0.418T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 - 8.71T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 8.71T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 0.274T + 67T^{2} \) |
| 71 | \( 1 - 8.24T + 71T^{2} \) |
| 73 | \( 1 + 2.72T + 73T^{2} \) |
| 79 | \( 1 + 3.72T + 79T^{2} \) |
| 83 | \( 1 - 0.837T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.761111523701017115040390572138, −7.67158585205074860937548910784, −7.14208605838425837466414130166, −6.62259809577394000355972051180, −5.22068565628013172400924556724, −4.64103925736509730624681315209, −3.54083280757949451606219200167, −3.08721348432785519221188908026, −1.40820824234301246847772820692, 0,
1.40820824234301246847772820692, 3.08721348432785519221188908026, 3.54083280757949451606219200167, 4.64103925736509730624681315209, 5.22068565628013172400924556724, 6.62259809577394000355972051180, 7.14208605838425837466414130166, 7.67158585205074860937548910784, 8.761111523701017115040390572138
