| L(s) = 1 | + 5-s − 2.56·7-s + 2·11-s − 3.56·13-s − 1.43·17-s − 2·19-s + 23-s + 25-s + 8.12·29-s + 0.123·31-s − 2.56·35-s + 0.561·37-s − 4.12·41-s + 6.24·43-s + 8.68·47-s − 0.438·49-s − 8.56·53-s + 2·55-s + 1.43·59-s − 0.876·61-s − 3.56·65-s − 7.43·67-s + 5·71-s + 7.56·73-s − 5.12·77-s − 0.876·79-s − 7.68·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.968·7-s + 0.603·11-s − 0.987·13-s − 0.348·17-s − 0.458·19-s + 0.208·23-s + 0.200·25-s + 1.50·29-s + 0.0221·31-s − 0.432·35-s + 0.0923·37-s − 0.643·41-s + 0.952·43-s + 1.26·47-s − 0.0626·49-s − 1.17·53-s + 0.269·55-s + 0.187·59-s − 0.112·61-s − 0.441·65-s − 0.908·67-s + 0.593·71-s + 0.885·73-s − 0.583·77-s − 0.0986·79-s − 0.843·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 8.12T + 29T^{2} \) |
| 31 | \( 1 - 0.123T + 31T^{2} \) |
| 37 | \( 1 - 0.561T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 + 0.876T + 61T^{2} \) |
| 67 | \( 1 + 7.43T + 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 + 0.876T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 5.12T + 97T^{2} \) |
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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
−7.32068262392772474339333790453, −6.63101370125062127710620596072, −6.29613099680643569197451954347, −5.40312673080186153371872737751, −4.63209154965497340754286890649, −3.92149708989825484818130023771, −2.93304302354514397979791273453, −2.39190665924841724012892180954, −1.23035312768112465917143825422, 0,
1.23035312768112465917143825422, 2.39190665924841724012892180954, 2.93304302354514397979791273453, 3.92149708989825484818130023771, 4.63209154965497340754286890649, 5.40312673080186153371872737751, 6.29613099680643569197451954347, 6.63101370125062127710620596072, 7.32068262392772474339333790453
