L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s − 2·11-s − 12-s − 5·13-s + 2·14-s + 16-s + 5·17-s + 18-s − 2·21-s − 2·22-s − 6·23-s − 24-s − 5·26-s − 27-s + 2·28-s − 6·29-s − 3·31-s + 32-s + 2·33-s + 5·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.38·13-s + 0.534·14-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 0.436·21-s − 0.426·22-s − 1.25·23-s − 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.538·31-s + 0.176·32-s + 0.348·33-s + 0.857·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.58419351984687457667573140135, −7.21115344987385939687370875272, −6.08368930405113508729994491570, −5.52550728170357616626040395632, −4.96741492278671797872727372950, −4.29242354549796354995393061148, −3.36149960340154795739548495393, −2.35287115680912918605694221158, −1.53152076238487340869438914628, 0,
1.53152076238487340869438914628, 2.35287115680912918605694221158, 3.36149960340154795739548495393, 4.29242354549796354995393061148, 4.96741492278671797872727372950, 5.52550728170357616626040395632, 6.08368930405113508729994491570, 7.21115344987385939687370875272, 7.58419351984687457667573140135