L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s − 2·9-s + 2·10-s + 11-s + 12-s − 3·13-s − 14-s − 2·15-s + 16-s + 4·17-s + 2·18-s − 2·20-s + 21-s − 22-s − 7·23-s − 24-s − 25-s + 3·26-s − 5·27-s + 28-s − 4·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.970·17-s + 0.471·18-s − 0.447·20-s + 0.218·21-s − 0.213·22-s − 1.45·23-s − 0.204·24-s − 1/5·25-s + 0.588·26-s − 0.962·27-s + 0.188·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−9.600636563361992459398018375211, −8.483277895412604925293896970956, −8.016164991805059167256739335819, −7.46247601215283147246814584680, −6.30398323834491679796732814249, −5.21904092719584321096152271249, −3.95645937676677069341934733066, −3.05491774757589117795495158054, −1.83422339122646596725491937064, 0,
1.83422339122646596725491937064, 3.05491774757589117795495158054, 3.95645937676677069341934733066, 5.21904092719584321096152271249, 6.30398323834491679796732814249, 7.46247601215283147246814584680, 8.016164991805059167256739335819, 8.483277895412604925293896970956, 9.600636563361992459398018375211