| L(s) = 1 | + 5-s + 7-s + 4·11-s − 2·13-s + 6·17-s + 25-s + 6·29-s + 35-s − 6·37-s + 10·41-s − 12·47-s + 49-s + 6·53-s + 4·55-s − 4·59-s + 2·61-s − 2·65-s − 8·67-s − 4·71-s + 10·73-s + 4·77-s + 12·83-s + 6·85-s + 10·89-s − 2·91-s − 14·97-s − 10·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 1/5·25-s + 1.11·29-s + 0.169·35-s − 0.986·37-s + 1.56·41-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s − 0.474·71-s + 1.17·73-s + 0.455·77-s + 1.31·83-s + 0.650·85-s + 1.05·89-s − 0.209·91-s − 1.42·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.614354521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.614354521\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.218265115760027393292193549342, −7.53377257233798889290186226589, −6.75922298302738127194460043048, −6.09978540053710262811104372187, −5.31408736804311279699452502828, −4.61183098771910700532880364218, −3.71025096927059116689194413647, −2.87356633269285060103642759532, −1.78881561496237411313420919192, −0.946675498713093935978578515695,
0.946675498713093935978578515695, 1.78881561496237411313420919192, 2.87356633269285060103642759532, 3.71025096927059116689194413647, 4.61183098771910700532880364218, 5.31408736804311279699452502828, 6.09978540053710262811104372187, 6.75922298302738127194460043048, 7.53377257233798889290186226589, 8.218265115760027393292193549342
