| L(s) = 1 | + 3-s + 9-s + 13-s − 2·17-s − 4·19-s − 4·23-s + 27-s − 6·29-s + 8·31-s + 6·37-s + 39-s − 2·41-s − 4·43-s − 7·49-s − 2·51-s − 6·53-s − 4·57-s − 2·61-s − 8·67-s − 4·69-s − 6·73-s − 4·79-s + 81-s + 12·83-s − 6·87-s + 6·89-s + 8·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.277·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s − 49-s − 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.256·61-s − 0.977·67-s − 0.481·69-s − 0.702·73-s − 0.450·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + 0.635·89-s + 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.257273387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.257273387\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.98678852198887, −14.66941497366746, −14.00361011370497, −13.49865103327871, −13.05370680185735, −12.59876935831881, −11.83950109095164, −11.41211693718539, −10.78259193902851, −10.16942698290602, −9.742255694590774, −9.048664902665455, −8.605962985000364, −7.985742762117786, −7.605679098855739, −6.759207935749016, −6.255957950913210, −5.768596413522466, −4.650476871151049, −4.478624530517933, −3.582482799966235, −3.040713025172574, −2.156852663763661, −1.690200121551763, −0.5354784637577695,
0.5354784637577695, 1.690200121551763, 2.156852663763661, 3.040713025172574, 3.582482799966235, 4.478624530517933, 4.650476871151049, 5.768596413522466, 6.255957950913210, 6.759207935749016, 7.605679098855739, 7.985742762117786, 8.605962985000364, 9.048664902665455, 9.742255694590774, 10.16942698290602, 10.78259193902851, 11.41211693718539, 11.83950109095164, 12.59876935831881, 13.05370680185735, 13.49865103327871, 14.00361011370497, 14.66941497366746, 14.98678852198887
