L(s) = 1 | − 3-s + 7-s + 9-s − 4·11-s + 2·13-s − 2·17-s + 4·19-s − 21-s − 8·23-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 2·37-s − 2·39-s + 2·41-s − 12·43-s − 8·47-s + 49-s + 2·51-s − 6·53-s − 4·57-s − 4·59-s − 2·61-s + 63-s + 12·67-s + 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 1.66·23-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + 1.46·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.62040850760099379245531878262, −6.46379906616428050757597166601, −6.27178493939193215671751387011, −5.16979981466180292046816446076, −4.89781871737444044034655018901, −3.97535482997512009915790765961, −3.05388649955946964918871017037, −2.18028466559387375683274521349, −1.16998532112163288308882600680, 0,
1.16998532112163288308882600680, 2.18028466559387375683274521349, 3.05388649955946964918871017037, 3.97535482997512009915790765961, 4.89781871737444044034655018901, 5.16979981466180292046816446076, 6.27178493939193215671751387011, 6.46379906616428050757597166601, 7.62040850760099379245531878262