L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 8-s + 9-s + 3·10-s + 4·11-s + 12-s − 3·13-s − 3·15-s + 16-s − 18-s − 3·20-s − 4·22-s − 23-s − 24-s + 4·25-s + 3·26-s + 27-s + 29-s + 3·30-s + 2·31-s − 32-s + 4·33-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.774·15-s + 1/4·16-s − 0.235·18-s − 0.670·20-s − 0.852·22-s − 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.588·26-s + 0.192·27-s + 0.185·29-s + 0.547·30-s + 0.359·31-s − 0.176·32-s + 0.696·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−7.72984510005466926767635591493, −7.00682403300823758782944348276, −6.72930765159729369050500891741, −5.51847013989405402076733526708, −4.51403745497681300670901690347, −3.86658328874784466129529075099, −3.21575754090618482225674584293, −2.23074455193604380094750369526, −1.18108152534614209277844902025, 0,
1.18108152534614209277844902025, 2.23074455193604380094750369526, 3.21575754090618482225674584293, 3.86658328874784466129529075099, 4.51403745497681300670901690347, 5.51847013989405402076733526708, 6.72930765159729369050500891741, 7.00682403300823758782944348276, 7.72984510005466926767635591493