| L(s) = 1 | + 1.94·3-s + 1.48·5-s − 7-s + 0.765·9-s + 11-s + 13-s + 2.87·15-s − 5.79·17-s + 2.95·19-s − 1.94·21-s − 3.43·23-s − 2.80·25-s − 4.33·27-s − 2.27·29-s − 4.29·31-s + 1.94·33-s − 1.48·35-s − 7.63·37-s + 1.94·39-s − 3.26·41-s − 6.24·43-s + 1.13·45-s − 5.19·47-s + 49-s − 11.2·51-s + 14.1·53-s + 1.48·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 0.663·5-s − 0.377·7-s + 0.255·9-s + 0.301·11-s + 0.277·13-s + 0.742·15-s − 1.40·17-s + 0.678·19-s − 0.423·21-s − 0.715·23-s − 0.560·25-s − 0.834·27-s − 0.423·29-s − 0.771·31-s + 0.337·33-s − 0.250·35-s − 1.25·37-s + 0.310·39-s − 0.510·41-s − 0.952·43-s + 0.169·45-s − 0.758·47-s + 0.142·49-s − 1.57·51-s + 1.94·53-s + 0.199·55-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 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 - 1.48T + 5T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 - 2.95T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 + 7.63T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 + 0.168T + 59T^{2} \) |
| 61 | \( 1 + 1.15T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 9.12T + 73T^{2} \) |
| 79 | \( 1 + 3.90T + 79T^{2} \) |
| 83 | \( 1 + 1.99T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
| 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
−7.46013011026792242994359940734, −6.93569101918329020162900867671, −6.06898939821666319594285871390, −5.54790194162172823701305838181, −4.51646660859508527641408036367, −3.68883759581093773933177044480, −3.13713166117487649157188235924, −2.15350763867920484572756021275, −1.69431119173604730834224934210, 0,
1.69431119173604730834224934210, 2.15350763867920484572756021275, 3.13713166117487649157188235924, 3.68883759581093773933177044480, 4.51646660859508527641408036367, 5.54790194162172823701305838181, 6.06898939821666319594285871390, 6.93569101918329020162900867671, 7.46013011026792242994359940734
