| L(s) = 1 | − 1.55·3-s − 0.910·5-s − 7-s − 0.592·9-s − 11-s − 13-s + 1.41·15-s + 6.01·17-s − 0.663·19-s + 1.55·21-s + 5.85·23-s − 4.17·25-s + 5.57·27-s − 3.16·29-s − 9.74·31-s + 1.55·33-s + 0.910·35-s − 2.95·37-s + 1.55·39-s − 3.08·41-s + 5.25·43-s + 0.539·45-s + 6.11·47-s + 49-s − 9.33·51-s + 9.32·53-s + 0.910·55-s + ⋯ |
| L(s) = 1 | − 0.895·3-s − 0.407·5-s − 0.377·7-s − 0.197·9-s − 0.301·11-s − 0.277·13-s + 0.364·15-s + 1.45·17-s − 0.152·19-s + 0.338·21-s + 1.22·23-s − 0.834·25-s + 1.07·27-s − 0.586·29-s − 1.74·31-s + 0.270·33-s + 0.153·35-s − 0.485·37-s + 0.248·39-s − 0.481·41-s + 0.802·43-s + 0.0803·45-s + 0.892·47-s + 0.142·49-s − 1.30·51-s + 1.28·53-s + 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 + 0.910T + 5T^{2} \) |
| 17 | \( 1 - 6.01T + 17T^{2} \) |
| 19 | \( 1 + 0.663T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 9.74T + 31T^{2} \) |
| 37 | \( 1 + 2.95T + 37T^{2} \) |
| 41 | \( 1 + 3.08T + 41T^{2} \) |
| 43 | \( 1 - 5.25T + 43T^{2} \) |
| 47 | \( 1 - 6.11T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 - 0.357T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 - 1.33T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 0.721T + 89T^{2} \) |
| 97 | \( 1 - 4.59T + 97T^{2} \) |
| 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
−7.39933686569091551993935142899, −6.86563692046361293934303334983, −5.86960756139028164265481290598, −5.51792274445486872192013960863, −4.89401448549340882173763539952, −3.81524355525294658388608278745, −3.26200035161576270028034101515, −2.23653456983946074040187151589, −0.968200407501423047824460775005, 0,
0.968200407501423047824460775005, 2.23653456983946074040187151589, 3.26200035161576270028034101515, 3.81524355525294658388608278745, 4.89401448549340882173763539952, 5.51792274445486872192013960863, 5.86960756139028164265481290598, 6.86563692046361293934303334983, 7.39933686569091551993935142899
