| L(s) = 1 | + 3-s − 5·7-s + 9-s + 5·11-s + 13-s − 3·17-s − 4·19-s − 5·21-s + 5·23-s + 27-s − 4·29-s + 5·33-s − 7·37-s + 39-s + 11·41-s − 12·43-s + 6·47-s + 18·49-s − 3·51-s + 53-s − 4·57-s + 12·59-s − 7·61-s − 5·63-s + 4·67-s + 5·69-s − 7·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.727·17-s − 0.917·19-s − 1.09·21-s + 1.04·23-s + 0.192·27-s − 0.742·29-s + 0.870·33-s − 1.15·37-s + 0.160·39-s + 1.71·41-s − 1.82·43-s + 0.875·47-s + 18/7·49-s − 0.420·51-s + 0.137·53-s − 0.529·57-s + 1.56·59-s − 0.896·61-s − 0.629·63-s + 0.488·67-s + 0.601·69-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.20097359490771831425027617382, −6.85877261362368570323372411334, −6.32184353209160291653618942462, −5.61087398737224947947179430942, −4.34149699735162375887751579818, −3.83798698840120102530072050724, −3.19117005113441443575831995119, −2.39657922240394678704191084242, −1.29439956450967371194308707026, 0,
1.29439956450967371194308707026, 2.39657922240394678704191084242, 3.19117005113441443575831995119, 3.83798698840120102530072050724, 4.34149699735162375887751579818, 5.61087398737224947947179430942, 6.32184353209160291653618942462, 6.85877261362368570323372411334, 7.20097359490771831425027617382
