| L(s) = 1 | − 0.488·2-s + 2.60·3-s − 1.76·4-s + 1.92·5-s − 1.27·6-s − 0.259·7-s + 1.83·8-s + 3.78·9-s − 0.940·10-s − 4.58·12-s + 2.14·13-s + 0.126·14-s + 5.02·15-s + 2.62·16-s − 4.30·17-s − 1.84·18-s − 7.44·19-s − 3.39·20-s − 0.675·21-s + 3.82·23-s + 4.78·24-s − 1.28·25-s − 1.04·26-s + 2.05·27-s + 0.456·28-s + 3.58·29-s − 2.45·30-s + ⋯ |
| L(s) = 1 | − 0.345·2-s + 1.50·3-s − 0.880·4-s + 0.861·5-s − 0.519·6-s − 0.0979·7-s + 0.649·8-s + 1.26·9-s − 0.297·10-s − 1.32·12-s + 0.596·13-s + 0.0338·14-s + 1.29·15-s + 0.656·16-s − 1.04·17-s − 0.435·18-s − 1.70·19-s − 0.759·20-s − 0.147·21-s + 0.798·23-s + 0.976·24-s − 0.257·25-s − 0.205·26-s + 0.394·27-s + 0.0862·28-s + 0.665·29-s − 0.447·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.488T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 + 0.259T + 7T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 + 7.44T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 8.02T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.99T + 59T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.24T + 73T^{2} \) |
| 79 | \( 1 - 9.01T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
| 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
−7.978543553771720531255692172186, −6.84387075389413440274055347907, −6.42378592626196705663826745757, −5.26900603084171812125720568683, −4.63311266991045617164880854091, −3.74654520613733517887718367824, −3.21562568142124754583579386137, −2.01441889999056453257087484760, −1.68878085480623405791736926021, 0,
1.68878085480623405791736926021, 2.01441889999056453257087484760, 3.21562568142124754583579386137, 3.74654520613733517887718367824, 4.63311266991045617164880854091, 5.26900603084171812125720568683, 6.42378592626196705663826745757, 6.84387075389413440274055347907, 7.978543553771720531255692172186
