| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s − 4·11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 4·19-s − 20-s − 21-s + 4·22-s − 4·23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.91285347904665, −14.45998480257915, −13.72070815423300, −13.44752748932317, −12.75884930154209, −12.25754549217342, −11.79469021797898, −11.10711179035104, −10.57304065039947, −10.15833998868912, −9.648134599550287, −9.029729845611194, −8.542719204809034, −7.991107726106329, −7.594912729246958, −7.110036124426088, −6.372432889013523, −5.848105522497146, −5.016156536454985, −4.513818365205123, −3.510591870642355, −3.204778139303824, −2.472115930343356, −1.810315467429763, −0.8543415433238767, 0,
0.8543415433238767, 1.810315467429763, 2.472115930343356, 3.204778139303824, 3.510591870642355, 4.513818365205123, 5.016156536454985, 5.848105522497146, 6.372432889013523, 7.110036124426088, 7.594912729246958, 7.991107726106329, 8.542719204809034, 9.029729845611194, 9.648134599550287, 10.15833998868912, 10.57304065039947, 11.10711179035104, 11.79469021797898, 12.25754549217342, 12.75884930154209, 13.44752748932317, 13.72070815423300, 14.45998480257915, 14.91285347904665
