| L(s) = 1 | − 5-s + 13-s + 2·17-s + 2·19-s − 6·23-s + 25-s + 2·29-s + 6·31-s + 6·37-s + 10·41-s − 4·43-s + 12·53-s + 12·59-s − 65-s + 16·67-s − 8·71-s − 14·73-s + 4·83-s − 2·85-s − 6·89-s − 2·95-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.277·13-s + 0.485·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s + 1.07·31-s + 0.986·37-s + 1.56·41-s − 0.609·43-s + 1.64·53-s + 1.56·59-s − 0.124·65-s + 1.95·67-s − 0.949·71-s − 1.63·73-s + 0.439·83-s − 0.216·85-s − 0.635·89-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.00718979120097, −12.80192038918319, −12.11649853871615, −11.70768014095078, −11.51137823586668, −10.87201560714915, −10.27124014306387, −9.943018648499488, −9.570339082742328, −8.780485729261275, −8.501887480662750, −7.919715173323539, −7.595180807785697, −7.031171847358016, −6.463439372745923, −5.955080317108102, −5.532354468295540, −4.914551494322602, −4.277278841074239, −3.918693101060392, −3.394014532489863, −2.562888994344379, −2.360595344873560, −1.253415037981705, −0.9064833492757754, 0,
0.9064833492757754, 1.253415037981705, 2.360595344873560, 2.562888994344379, 3.394014532489863, 3.918693101060392, 4.277278841074239, 4.914551494322602, 5.532354468295540, 5.955080317108102, 6.463439372745923, 7.031171847358016, 7.595180807785697, 7.919715173323539, 8.501887480662750, 8.780485729261275, 9.570339082742328, 9.943018648499488, 10.27124014306387, 10.87201560714915, 11.51137823586668, 11.70768014095078, 12.11649853871615, 12.80192038918319, 13.00718979120097
