| L(s) = 1 | − 5-s − 3·9-s − 11-s − 2·13-s + 6·17-s − 4·19-s − 2·23-s + 25-s − 2·29-s + 10·31-s + 2·37-s + 12·41-s + 3·45-s − 6·53-s + 55-s − 6·59-s + 2·65-s + 2·67-s − 8·71-s + 10·73-s + 9·81-s − 4·83-s − 6·85-s + 6·89-s + 4·95-s + 2·97-s + 3·99-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.417·23-s + 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.328·37-s + 1.87·41-s + 0.447·45-s − 0.824·53-s + 0.134·55-s − 0.781·59-s + 0.248·65-s + 0.244·67-s − 0.949·71-s + 1.17·73-s + 81-s − 0.439·83-s − 0.650·85-s + 0.635·89-s + 0.410·95-s + 0.203·97-s + 0.301·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.84110741695164, −16.28337889185128, −15.69677455278136, −15.01800749640611, −14.53354293380148, −14.10684971783130, −13.45162037128635, −12.53992696462915, −12.33162229124673, −11.60024785761678, −11.10344920505039, −10.40112068502405, −9.846976992965181, −9.153814547184172, −8.438318595985185, −7.836156182926311, −7.541410375584554, −6.409171210745232, −6.000432423038675, −5.204888733763600, −4.538734770034809, −3.735226702361592, −2.908786028116562, −2.360175086386414, −1.054467272227588, 0,
1.054467272227588, 2.360175086386414, 2.908786028116562, 3.735226702361592, 4.538734770034809, 5.204888733763600, 6.000432423038675, 6.409171210745232, 7.541410375584554, 7.836156182926311, 8.438318595985185, 9.153814547184172, 9.846976992965181, 10.40112068502405, 11.10344920505039, 11.60024785761678, 12.33162229124673, 12.53992696462915, 13.45162037128635, 14.10684971783130, 14.53354293380148, 15.01800749640611, 15.69677455278136, 16.28337889185128, 16.84110741695164
