| L(s) = 1 | + 1.69·3-s − 3.84·5-s − 7-s − 0.114·9-s − 11-s − 13-s − 6.52·15-s − 3.95·17-s − 0.644·19-s − 1.69·21-s + 9.08·23-s + 9.76·25-s − 5.29·27-s + 5.91·29-s − 1.51·31-s − 1.69·33-s + 3.84·35-s − 4.94·37-s − 1.69·39-s − 4.71·41-s + 6.12·43-s + 0.440·45-s − 11.9·47-s + 49-s − 6.71·51-s + 3.05·53-s + 3.84·55-s + ⋯ |
| L(s) = 1 | + 0.980·3-s − 1.71·5-s − 0.377·7-s − 0.0382·9-s − 0.301·11-s − 0.277·13-s − 1.68·15-s − 0.959·17-s − 0.147·19-s − 0.370·21-s + 1.89·23-s + 1.95·25-s − 1.01·27-s + 1.09·29-s − 0.272·31-s − 0.295·33-s + 0.649·35-s − 0.812·37-s − 0.271·39-s − 0.736·41-s + 0.934·43-s + 0.0656·45-s − 1.74·47-s + 0.142·49-s − 0.940·51-s + 0.420·53-s + 0.518·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.275909735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.275909735\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 + 3.84T + 5T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 + 0.644T + 19T^{2} \) |
| 23 | \( 1 - 9.08T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 + 4.71T + 41T^{2} \) |
| 43 | \( 1 - 6.12T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 3.05T + 53T^{2} \) |
| 59 | \( 1 - 2.39T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 8.90T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 8.07T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.293312842494329532765601271081, −7.984779041878348840307621001035, −7.01619440557207108707861523337, −6.67183492378270345586692085363, −5.18961603961769082074758352583, −4.55639932842581252373489954448, −3.58264365099289125703430149251, −3.16728857188006810621265246670, −2.27297012882301674633631324534, −0.58896781279062181126610340364,
0.58896781279062181126610340364, 2.27297012882301674633631324534, 3.16728857188006810621265246670, 3.58264365099289125703430149251, 4.55639932842581252373489954448, 5.18961603961769082074758352583, 6.67183492378270345586692085363, 7.01619440557207108707861523337, 7.984779041878348840307621001035, 8.293312842494329532765601271081
