| L(s) = 1 | − 3-s + 2·7-s + 9-s + 2·11-s + 2·13-s + 2·17-s − 2·19-s − 2·21-s + 2·23-s − 27-s − 6·29-s − 4·31-s − 2·33-s − 2·37-s − 2·39-s − 10·41-s − 8·43-s + 2·47-s − 3·49-s − 2·51-s − 6·53-s + 2·57-s − 2·59-s − 10·61-s + 2·63-s + 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.485·17-s − 0.458·19-s − 0.436·21-s + 0.417·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.328·37-s − 0.320·39-s − 1.56·41-s − 1.21·43-s + 0.291·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s + 0.264·57-s − 0.260·59-s − 1.28·61-s + 0.251·63-s + 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.19430276880221896016243975898, −6.72066738243678890496069233290, −5.91249932971871937041951934394, −5.31168806574047594359994375210, −4.66513590350734753078312675018, −3.86226643015954962930877526332, −3.18889135941031064946096699950, −1.84054060731714946794849913562, −1.35947901115361627131306812868, 0,
1.35947901115361627131306812868, 1.84054060731714946794849913562, 3.18889135941031064946096699950, 3.86226643015954962930877526332, 4.66513590350734753078312675018, 5.31168806574047594359994375210, 5.91249932971871937041951934394, 6.72066738243678890496069233290, 7.19430276880221896016243975898
