| 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 − 5·17-s − 18-s − 8·19-s − 20-s + 4·22-s − 23-s + 24-s + 25-s − 2·26-s − 27-s − 29-s − 30-s + 7·31-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 − 1.21·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.852·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.185·29-s − 0.182·30-s + 1.25·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.53203559089172, −15.25924981514169, −14.77298440472969, −13.74264520055706, −13.27990713896108, −12.94987353768566, −12.16192266713248, −11.75669665977094, −11.18108446818021, −10.63071524625557, −10.32267545365717, −9.872536380863430, −8.783717428387284, −8.475548152651760, −8.236407432565878, −7.229865727754278, −6.849813959164937, −6.342238700807521, −5.628278946709480, −4.981487124376526, −4.319016825239201, −3.714609648575222, −2.729744035102619, −2.159927997901131, −1.330820280960267, 0, 0,
1.330820280960267, 2.159927997901131, 2.729744035102619, 3.714609648575222, 4.319016825239201, 4.981487124376526, 5.628278946709480, 6.342238700807521, 6.849813959164937, 7.229865727754278, 8.236407432565878, 8.475548152651760, 8.783717428387284, 9.872536380863430, 10.32267545365717, 10.63071524625557, 11.18108446818021, 11.75669665977094, 12.16192266713248, 12.94987353768566, 13.27990713896108, 13.74264520055706, 14.77298440472969, 15.25924981514169, 15.53203559089172
