| L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s + 6·13-s − 2·17-s − 4·19-s − 21-s + 8·23-s + 27-s + 2·29-s − 4·33-s − 10·37-s + 6·39-s − 6·41-s + 4·43-s + 49-s − 2·51-s + 6·53-s − 4·57-s + 4·59-s − 6·61-s − 63-s − 4·67-s + 8·69-s − 8·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 1.64·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.125·63-s − 0.488·67-s + 0.963·69-s − 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.34105036915970, −14.87768611911133, −14.08766471385324, −13.57641829089569, −13.13469904945172, −12.94507703884195, −12.21072167586817, −11.46521871152165, −10.87448627020241, −10.44633522957937, −10.12520659042612, −9.049411237826015, −8.784709845346452, −8.464907877941447, −7.663293919055153, −7.087716052136132, −6.530192601434237, −5.927454983175663, −5.216469105226557, −4.608652977451166, −3.846631629274388, −3.246437768248460, −2.712034310435650, −1.905385633567923, −1.087770642663804, 0,
1.087770642663804, 1.905385633567923, 2.712034310435650, 3.246437768248460, 3.846631629274388, 4.608652977451166, 5.216469105226557, 5.927454983175663, 6.530192601434237, 7.087716052136132, 7.663293919055153, 8.464907877941447, 8.784709845346452, 9.049411237826015, 10.12520659042612, 10.44633522957937, 10.87448627020241, 11.46521871152165, 12.21072167586817, 12.94507703884195, 13.13469904945172, 13.57641829089569, 14.08766471385324, 14.87768611911133, 15.34105036915970
