L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s − 2·9-s + 10-s + 5·11-s − 12-s + 13-s + 2·14-s + 15-s + 16-s + 2·17-s + 2·18-s − 4·19-s − 20-s + 2·21-s − 5·22-s − 6·23-s + 24-s − 4·25-s − 26-s + 5·27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.471·18-s − 0.917·19-s − 0.223·20-s + 0.436·21-s − 1.06·22-s − 1.25·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.962·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6338117364\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6338117364\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.392438849219341735831503216477, −8.586275555206076946225421803593, −7.909261382378427002878419322856, −6.82120008949079200459227101857, −6.26762861701016783232021065979, −5.61555428161067181084448072737, −4.13430945936064174658517710222, −3.46580636331507380814960440849, −2.08026204628831579531853388934, −0.61439046057345923085639409537,
0.61439046057345923085639409537, 2.08026204628831579531853388934, 3.46580636331507380814960440849, 4.13430945936064174658517710222, 5.61555428161067181084448072737, 6.26762861701016783232021065979, 6.82120008949079200459227101857, 7.909261382378427002878419322856, 8.586275555206076946225421803593, 9.392438849219341735831503216477