L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s − 11-s − 12-s + 13-s + 3·14-s + 16-s − 17-s + 18-s − 8·19-s − 3·21-s − 22-s + 2·23-s − 24-s + 26-s − 27-s + 3·28-s + 6·29-s + 32-s + 33-s − 34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s − 0.654·21-s − 0.213·22-s + 0.417·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.566·28-s + 1.11·29-s + 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.80078421944119, −12.33478072349064, −11.85712873045181, −11.32527836770578, −11.04286711113585, −10.55998338128802, −10.40423001816796, −9.659927319622287, −8.989451975376643, −8.570986029710049, −8.082705965913411, −7.731291714224181, −7.090676742673235, −6.521435854887980, −6.281351187874459, −5.771530638535563, −4.968708210034290, −4.824169410567899, −4.514600686794760, −3.768585795016899, −3.311993380769512, −2.534663493070111, −1.984405894286797, −1.588959806254405, −0.8236926772815966, 0,
0.8236926772815966, 1.588959806254405, 1.984405894286797, 2.534663493070111, 3.311993380769512, 3.768585795016899, 4.514600686794760, 4.824169410567899, 4.968708210034290, 5.771530638535563, 6.281351187874459, 6.521435854887980, 7.090676742673235, 7.731291714224181, 8.082705965913411, 8.570986029710049, 8.989451975376643, 9.659927319622287, 10.40423001816796, 10.55998338128802, 11.04286711113585, 11.32527836770578, 11.85712873045181, 12.33478072349064, 12.80078421944119