| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s + 7-s − 8-s + 9-s − 10-s + 2·12-s + 6·13-s − 14-s + 2·15-s + 16-s − 18-s + 7·19-s + 20-s + 2·21-s + 3·23-s − 2·24-s + 25-s − 6·26-s − 4·27-s + 28-s − 6·29-s − 2·30-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 1.66·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.60·19-s + 0.223·20-s + 0.436·21-s + 0.625·23-s − 0.408·24-s + 1/5·25-s − 1.17·26-s − 0.769·27-s + 0.188·28-s − 1.11·29-s − 0.365·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.86413961889443, −13.55755819072831, −13.35500658075735, −12.59128652668843, −11.91395843037979, −11.35609873428926, −11.12199114728807, −10.43989288867972, −9.898008432070546, −9.401974714820918, −9.049022428898150, −8.458010404601487, −8.253472470092796, −7.661788283251468, −7.066867521351512, −6.603514746558782, −5.886463109199868, −5.356577129976583, −4.863567242636240, −3.670504131313223, −3.531420308523835, −2.970996822812250, −2.189611127405338, −1.485299368732042, −1.262533610086548, 0,
1.262533610086548, 1.485299368732042, 2.189611127405338, 2.970996822812250, 3.531420308523835, 3.670504131313223, 4.863567242636240, 5.356577129976583, 5.886463109199868, 6.603514746558782, 7.066867521351512, 7.661788283251468, 8.253472470092796, 8.458010404601487, 9.049022428898150, 9.401974714820918, 9.898008432070546, 10.43989288867972, 11.12199114728807, 11.35609873428926, 11.91395843037979, 12.59128652668843, 13.35500658075735, 13.55755819072831, 13.86413961889443
