| L(s) = 1 | + 2·7-s − 2·11-s + 13-s − 4·17-s + 3·19-s + 6·23-s + 29-s − 37-s + 5·41-s − 4·43-s + 3·47-s − 3·49-s − 7·53-s − 2·59-s − 10·61-s + 9·67-s + 71-s − 4·73-s − 4·77-s + 79-s − 10·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.603·11-s + 0.277·13-s − 0.970·17-s + 0.688·19-s + 1.25·23-s + 0.185·29-s − 0.164·37-s + 0.780·41-s − 0.609·43-s + 0.437·47-s − 3/7·49-s − 0.961·53-s − 0.260·59-s − 1.28·61-s + 1.09·67-s + 0.118·71-s − 0.468·73-s − 0.455·77-s + 0.112·79-s − 1.05·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.97756671584690, −14.20720646888776, −13.91127871894012, −13.33557297318674, −12.78268601496782, −12.41487748508592, −11.55543287950904, −11.23197316975237, −10.85954040739946, −10.24263854759533, −9.609302700548225, −8.995839909375857, −8.620259403218256, −7.895396889806942, −7.560484514191200, −6.846147007546584, −6.342619689316247, −5.577603141684657, −5.013025293198993, −4.628739376804476, −3.888734412127593, −3.078502296547716, −2.563760982846118, −1.708223730025643, −1.059454820803457, 0,
1.059454820803457, 1.708223730025643, 2.563760982846118, 3.078502296547716, 3.888734412127593, 4.628739376804476, 5.013025293198993, 5.577603141684657, 6.342619689316247, 6.846147007546584, 7.560484514191200, 7.895396889806942, 8.620259403218256, 8.995839909375857, 9.609302700548225, 10.24263854759533, 10.85954040739946, 11.23197316975237, 11.55543287950904, 12.41487748508592, 12.78268601496782, 13.33557297318674, 13.91127871894012, 14.20720646888776, 14.97756671584690
