L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s − 4·7-s + 8-s + 9-s + 4·10-s + 11-s − 12-s − 2·13-s − 4·14-s − 4·15-s + 16-s + 6·17-s + 18-s − 4·19-s + 4·20-s + 4·21-s + 22-s + 6·23-s − 24-s + 11·25-s − 2·26-s − 27-s − 4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s − 1.03·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.894·20-s + 0.872·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.127196644\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.127196644\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.77686437364902, −13.19635916536081, −12.97728551527894, −12.51773654212981, −12.16648201125462, −11.48711167549111, −10.77906781071174, −10.25743285279725, −9.997235952725593, −9.628556893183768, −9.029481406068362, −8.491396989160491, −7.489491726510276, −6.959429708480402, −6.465494912966537, −6.233245793962873, −5.627096132730385, −5.258286188619439, −4.675121599107895, −3.901865950252794, −3.186696219447854, −2.682034032414576, −2.203783560993155, −1.240110105376726, −0.7123539982812442,
0.7123539982812442, 1.240110105376726, 2.203783560993155, 2.682034032414576, 3.186696219447854, 3.901865950252794, 4.675121599107895, 5.258286188619439, 5.627096132730385, 6.233245793962873, 6.465494912966537, 6.959429708480402, 7.489491726510276, 8.491396989160491, 9.029481406068362, 9.628556893183768, 9.997235952725593, 10.25743285279725, 10.77906781071174, 11.48711167549111, 12.16648201125462, 12.51773654212981, 12.97728551527894, 13.19635916536081, 13.77686437364902