| L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 8-s + 1.41·10-s − 4·11-s + 4.24·13-s + 16-s + 7.07·17-s − 5.65·19-s − 1.41·20-s + 4·22-s − 8·23-s − 2.99·25-s − 4.24·26-s − 2·29-s − 32-s − 7.07·34-s + 4·37-s + 5.65·38-s + 1.41·40-s − 9.89·41-s − 4·43-s − 4·44-s + 8·46-s − 5.65·47-s + 2.99·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.632·5-s − 0.353·8-s + 0.447·10-s − 1.20·11-s + 1.17·13-s + 0.250·16-s + 1.71·17-s − 1.29·19-s − 0.316·20-s + 0.852·22-s − 1.66·23-s − 0.599·25-s − 0.832·26-s − 0.371·29-s − 0.176·32-s − 1.21·34-s + 0.657·37-s + 0.917·38-s + 0.223·40-s − 1.54·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s − 0.825·47-s + 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
| 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
−10.01720571963695241720881608827, −8.589941487088499416010561886341, −8.118049983618376092738830727856, −7.50287723561590050483594281799, −6.27011586922826886014853818576, −5.52792116357709787421509266144, −4.11473716940554774137568040594, −3.15874985183319832516167178509, −1.73758245242588432787724477923, 0,
1.73758245242588432787724477923, 3.15874985183319832516167178509, 4.11473716940554774137568040594, 5.52792116357709787421509266144, 6.27011586922826886014853818576, 7.50287723561590050483594281799, 8.118049983618376092738830727856, 8.589941487088499416010561886341, 10.01720571963695241720881608827
