| L(s) = 1 | − 4·7-s − 5·11-s − 13-s − 17-s − 7·19-s − 4·23-s − 2·31-s + 8·37-s − 7·41-s − 12·43-s + 9·49-s + 10·53-s − 4·59-s + 8·61-s + 11·67-s − 12·71-s − 7·73-s + 20·77-s + 9·83-s − 9·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.50·11-s − 0.277·13-s − 0.242·17-s − 1.60·19-s − 0.834·23-s − 0.359·31-s + 1.31·37-s − 1.09·41-s − 1.82·43-s + 9/7·49-s + 1.37·53-s − 0.520·59-s + 1.02·61-s + 1.34·67-s − 1.42·71-s − 0.819·73-s + 2.27·77-s + 0.987·83-s − 0.953·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.88167946675023, −13.42925039915253, −12.98906854228751, −12.81708312944875, −12.26828777540471, −11.59652692279890, −11.12001599140478, −10.34396530364201, −10.12253233326469, −9.880074192791498, −9.036371569716941, −8.580837648085934, −8.103983782677545, −7.503634106667965, −6.888975740468545, −6.466876446613798, −5.947426843261020, −5.421760185419251, −4.751382895805543, −4.158796825236948, −3.518638517492182, −2.945492301951074, −2.358469861113387, −1.875803202040539, −0.5320732979050788, 0,
0.5320732979050788, 1.875803202040539, 2.358469861113387, 2.945492301951074, 3.518638517492182, 4.158796825236948, 4.751382895805543, 5.421760185419251, 5.947426843261020, 6.466876446613798, 6.888975740468545, 7.503634106667965, 8.103983782677545, 8.580837648085934, 9.036371569716941, 9.880074192791498, 10.12253233326469, 10.34396530364201, 11.12001599140478, 11.59652692279890, 12.26828777540471, 12.81708312944875, 12.98906854228751, 13.42925039915253, 13.88167946675023
