L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 4·11-s + 12-s − 13-s − 15-s + 16-s − 17-s − 18-s + 4·19-s − 20-s + 4·22-s − 24-s + 25-s + 26-s + 27-s − 2·29-s + 30-s − 32-s − 4·33-s + 34-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 − 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.696·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.84958453078472, −12.31754459943642, −11.96494634463561, −11.27451996150087, −11.05396101902033, −10.51951658287088, −9.968607174586645, −9.662587883572134, −9.237906148980722, −8.591819628620595, −8.244143890714218, −7.847476306799845, −7.383925829896520, −7.085456157883486, −6.451535750791905, −5.833322373381761, −5.293676265190313, −4.805301353493589, −4.262194830004158, −3.542943506371771, −3.047402666451132, −2.694592170648061, −2.002950591835859, −1.470809671428419, −0.6518701432807705, 0,
0.6518701432807705, 1.470809671428419, 2.002950591835859, 2.694592170648061, 3.047402666451132, 3.542943506371771, 4.262194830004158, 4.805301353493589, 5.293676265190313, 5.833322373381761, 6.451535750791905, 7.085456157883486, 7.383925829896520, 7.847476306799845, 8.244143890714218, 8.591819628620595, 9.237906148980722, 9.662587883572134, 9.968607174586645, 10.51951658287088, 11.05396101902033, 11.27451996150087, 11.96494634463561, 12.31754459943642, 12.84958453078472