L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 3·11-s − 12-s + 6·13-s − 15-s + 16-s − 18-s − 19-s + 20-s + 3·22-s + 23-s + 24-s + 25-s − 6·26-s − 27-s − 2·29-s + 30-s − 32-s + 3·33-s + 36-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 − 0.904·11-s − 0.288·12-s + 1.66·13-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.639·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s + 0.522·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.071044415\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.071044415\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.06298183440223, −12.85410928637651, −12.32890151315647, −11.63073868001498, −11.16108768048197, −10.85388958999417, −10.51054836939284, −9.955911056378724, −9.422219387442390, −8.982384533088320, −8.450885478365088, −7.973081198003631, −7.529621835236271, −6.848392206113044, −6.376233585465247, −6.005129282643051, −5.411027826604814, −5.052764769145551, −4.203701949415131, −3.647588845954305, −3.070015103609086, −2.211487472405412, −1.890371133858681, −0.9002645440332040, −0.6310912361537244,
0.6310912361537244, 0.9002645440332040, 1.890371133858681, 2.211487472405412, 3.070015103609086, 3.647588845954305, 4.203701949415131, 5.052764769145551, 5.411027826604814, 6.005129282643051, 6.376233585465247, 6.848392206113044, 7.529621835236271, 7.973081198003631, 8.450885478365088, 8.982384533088320, 9.422219387442390, 9.955911056378724, 10.51054836939284, 10.85388958999417, 11.16108768048197, 11.63073868001498, 12.32890151315647, 12.85410928637651, 13.06298183440223