L(s) = 1 | + 0.297·2-s − 3-s − 1.91·4-s + 0.750·5-s − 0.297·6-s + 4.45·7-s − 1.16·8-s + 9-s + 0.223·10-s − 1.72·11-s + 1.91·12-s − 1.50·13-s + 1.32·14-s − 0.750·15-s + 3.47·16-s − 17-s + 0.297·18-s + 4.02·19-s − 1.43·20-s − 4.45·21-s − 0.514·22-s − 4.55·23-s + 1.16·24-s − 4.43·25-s − 0.448·26-s − 27-s − 8.51·28-s + ⋯ |
L(s) = 1 | + 0.210·2-s − 0.577·3-s − 0.955·4-s + 0.335·5-s − 0.121·6-s + 1.68·7-s − 0.411·8-s + 0.333·9-s + 0.0706·10-s − 0.521·11-s + 0.551·12-s − 0.417·13-s + 0.354·14-s − 0.193·15-s + 0.869·16-s − 0.242·17-s + 0.0701·18-s + 0.923·19-s − 0.320·20-s − 0.972·21-s − 0.109·22-s − 0.949·23-s + 0.237·24-s − 0.887·25-s − 0.0879·26-s − 0.192·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.644232057\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.644232057\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 - 0.297T + 2T^{2} \) |
| 5 | \( 1 - 0.750T + 5T^{2} \) |
| 7 | \( 1 - 4.45T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + 1.50T + 13T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 23 | \( 1 + 4.55T + 23T^{2} \) |
| 29 | \( 1 + 7.42T + 29T^{2} \) |
| 31 | \( 1 - 8.04T + 31T^{2} \) |
| 37 | \( 1 - 7.09T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 - 3.45T + 43T^{2} \) |
| 47 | \( 1 + 2.50T + 47T^{2} \) |
| 53 | \( 1 - 0.595T + 53T^{2} \) |
| 59 | \( 1 - 4.69T + 59T^{2} \) |
| 61 | \( 1 + 1.27T + 61T^{2} \) |
| 67 | \( 1 + 1.15T + 67T^{2} \) |
| 71 | \( 1 + 3.42T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 6.05T + 79T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 - 8.95T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−8.005759474018766178252391578220, −7.32144206566656255111032908560, −6.15350843791374727324139924104, −5.54125040067848405380745314379, −5.08505948745778316643745370235, −4.44296786604012449356903375761, −3.85458970828970926098217868501, −2.56139065719753582730817316067, −1.67595142581749130735348866509, −0.66062066723352812578130837749,
0.66062066723352812578130837749, 1.67595142581749130735348866509, 2.56139065719753582730817316067, 3.85458970828970926098217868501, 4.44296786604012449356903375761, 5.08505948745778316643745370235, 5.54125040067848405380745314379, 6.15350843791374727324139924104, 7.32144206566656255111032908560, 8.005759474018766178252391578220