| L(s) = 1 | − 5-s − 4·7-s + 11-s − 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s + 4·35-s + 10·37-s + 6·41-s − 12·43-s − 4·47-s + 9·49-s − 6·53-s − 55-s + 4·59-s − 10·61-s + 2·65-s − 12·67-s − 4·71-s + 10·73-s − 4·77-s − 4·79-s − 4·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.676·35-s + 1.64·37-s + 0.937·41-s − 1.82·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.134·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s − 1.46·67-s − 0.474·71-s + 1.17·73-s − 0.455·77-s − 0.450·79-s − 0.439·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8475412809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8475412809\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.15160669828861, −14.54273576553096, −14.01101539471165, −13.38344012864213, −12.99455999547542, −12.39758161832243, −11.92691028470754, −11.49643262562686, −10.77769760678179, −10.04442284066163, −9.756195539392275, −9.264552255559270, −8.601901525904855, −7.907709213490569, −7.408033503371066, −6.717662729459441, −6.300514218691755, −5.772899674953788, −4.816365327512605, −4.354037546146419, −3.547991193153651, −3.038245213273823, −2.457260846674209, −1.348287096687241, −0.3604998947917234,
0.3604998947917234, 1.348287096687241, 2.457260846674209, 3.038245213273823, 3.547991193153651, 4.354037546146419, 4.816365327512605, 5.772899674953788, 6.300514218691755, 6.717662729459441, 7.408033503371066, 7.907709213490569, 8.601901525904855, 9.264552255559270, 9.756195539392275, 10.04442284066163, 10.77769760678179, 11.49643262562686, 11.92691028470754, 12.39758161832243, 12.99455999547542, 13.38344012864213, 14.01101539471165, 14.54273576553096, 15.15160669828861
