| L(s) = 1 | + 1.41·2-s − 2.41·3-s − 5-s − 3.41·6-s − 3.41·7-s − 2.82·8-s + 2.82·9-s − 1.41·10-s − 11-s + 2.24·13-s − 4.82·14-s + 2.41·15-s − 4.00·16-s + 3.41·17-s + 4·18-s − 19-s + 8.24·21-s − 1.41·22-s − 3·23-s + 6.82·24-s − 4·25-s + 3.17·26-s + 0.414·27-s − 6.24·29-s + 3.41·30-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 1.39·3-s − 0.447·5-s − 1.39·6-s − 1.29·7-s − 0.999·8-s + 0.942·9-s − 0.447·10-s − 0.301·11-s + 0.621·13-s − 1.29·14-s + 0.623·15-s − 1.00·16-s + 0.828·17-s + 0.942·18-s − 0.229·19-s + 1.79·21-s − 0.301·22-s − 0.625·23-s + 1.39·24-s − 0.800·25-s + 0.621·26-s + 0.0797·27-s − 1.15·29-s + 0.623·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 - 0.343T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 3.41T + 83T^{2} \) |
| 89 | \( 1 - 4.89T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
| 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
−12.07190025498865328151400289003, −11.27778035721523365343604018422, −10.17657948352155952925893447062, −9.121411903582605984216326523335, −7.46822210305757932839889624254, −5.99919994480041418957486446423, −5.85127016788906070010672256042, −4.37512198625136697918539487478, −3.33673779054846715421052009264, 0,
3.33673779054846715421052009264, 4.37512198625136697918539487478, 5.85127016788906070010672256042, 5.99919994480041418957486446423, 7.46822210305757932839889624254, 9.121411903582605984216326523335, 10.17657948352155952925893447062, 11.27778035721523365343604018422, 12.07190025498865328151400289003
