| L(s) = 1 | − 56·7-s − 156·11-s − 350·13-s + 786·17-s + 740·19-s + 2.37e3·23-s − 2.57e3·29-s − 4.57e3·31-s + 1.22e4·37-s + 1.02e4·41-s + 1.60e4·43-s + 864·47-s − 1.36e4·49-s − 1.76e4·53-s − 4.86e4·59-s − 3.37e4·61-s − 3.52e3·67-s − 3.82e4·71-s + 7.97e4·73-s + 8.73e3·77-s + 9.92e4·79-s − 2.22e4·83-s − 9.46e4·89-s + 1.96e4·91-s − 9.12e3·97-s + 7.15e4·101-s + 3.18e4·103-s + ⋯ |
| L(s) = 1 | − 0.431·7-s − 0.388·11-s − 0.574·13-s + 0.659·17-s + 0.470·19-s + 0.936·23-s − 0.568·29-s − 0.855·31-s + 1.46·37-s + 0.950·41-s + 1.32·43-s + 0.0570·47-s − 0.813·49-s − 0.863·53-s − 1.82·59-s − 1.16·61-s − 0.0959·67-s − 0.901·71-s + 1.75·73-s + 0.167·77-s + 1.78·79-s − 0.355·83-s − 1.26·89-s + 0.248·91-s − 0.0984·97-s + 0.698·101-s + 0.295·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 8 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 156 T + p^{5} T^{2} \) |
| 13 | \( 1 + 350 T + p^{5} T^{2} \) |
| 17 | \( 1 - 786 T + p^{5} T^{2} \) |
| 19 | \( 1 - 740 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2376 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2574 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4576 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12202 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10230 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16084 T + p^{5} T^{2} \) |
| 47 | \( 1 - 864 T + p^{5} T^{2} \) |
| 53 | \( 1 + 17658 T + p^{5} T^{2} \) |
| 59 | \( 1 + 48684 T + p^{5} T^{2} \) |
| 61 | \( 1 + 33778 T + p^{5} T^{2} \) |
| 67 | \( 1 + 3524 T + p^{5} T^{2} \) |
| 71 | \( 1 + 38280 T + p^{5} T^{2} \) |
| 73 | \( 1 - 79702 T + p^{5} T^{2} \) |
| 79 | \( 1 - 99248 T + p^{5} T^{2} \) |
| 83 | \( 1 + 22284 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94650 T + p^{5} T^{2} \) |
| 97 | \( 1 + 9122 T + p^{5} T^{2} \) |
| 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
−9.279990564992369883839886531650, −7.87399079724446829072709069006, −7.43068130212233025787600207492, −6.31187550977716602576461308976, −5.47067965429714877570860600491, −4.54576470706363622760301616667, −3.37152220773094652685193481327, −2.53124195595717288250630116092, −1.17942430704128889643965140892, 0,
1.17942430704128889643965140892, 2.53124195595717288250630116092, 3.37152220773094652685193481327, 4.54576470706363622760301616667, 5.47067965429714877570860600491, 6.31187550977716602576461308976, 7.43068130212233025787600207492, 7.87399079724446829072709069006, 9.279990564992369883839886531650
