| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 4·11-s + 12-s + 2·13-s − 15-s + 16-s − 6·17-s − 18-s − 2·19-s − 20-s + 4·22-s − 8·23-s − 24-s + 25-s − 2·26-s + 27-s + 6·29-s + 30-s − 6·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 1.07·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.15673559589123, −14.46256942419412, −13.90724332908519, −13.39082972546016, −12.92871445101838, −12.40215100351563, −11.83861788164116, −11.05677794262343, −10.93004280223034, −10.22143809558331, −9.820982709083207, −9.147195910105683, −8.567123461900556, −8.227468716365411, −7.808055199826874, −7.247493235542800, −6.446867712209210, −6.231434733276847, −5.293647525446978, −4.549017317664725, −4.152198989388268, −3.176071355680821, −2.869104053998657, −1.914021838380438, −1.582192914626740, 0, 0,
1.582192914626740, 1.914021838380438, 2.869104053998657, 3.176071355680821, 4.152198989388268, 4.549017317664725, 5.293647525446978, 6.231434733276847, 6.446867712209210, 7.247493235542800, 7.808055199826874, 8.227468716365411, 8.567123461900556, 9.147195910105683, 9.820982709083207, 10.22143809558331, 10.93004280223034, 11.05677794262343, 11.83861788164116, 12.40215100351563, 12.92871445101838, 13.39082972546016, 13.90724332908519, 14.46256942419412, 15.15673559589123
