| L(s) = 1 | − 8·4-s − 20·7-s + 70·13-s + 64·16-s + 56·19-s + 160·28-s + 308·31-s − 110·37-s + 520·43-s + 57·49-s − 560·52-s + 182·61-s − 512·64-s + 880·67-s − 1.19e3·73-s − 448·76-s + 884·79-s − 1.40e3·91-s + 1.33e3·97-s − 1.82e3·103-s − 646·109-s − 1.28e3·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.07·7-s + 1.49·13-s + 16-s + 0.676·19-s + 1.07·28-s + 1.78·31-s − 0.488·37-s + 1.84·43-s + 0.166·49-s − 1.49·52-s + 0.382·61-s − 64-s + 1.60·67-s − 1.90·73-s − 0.676·76-s + 1.25·79-s − 1.61·91-s + 1.39·97-s − 1.74·103-s − 0.567·109-s − 1.07·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.223521255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223521255\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 520 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 182 T + p^{3} T^{2} \) |
| 67 | \( 1 - 880 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 1330 T + p^{3} T^{2} \) |
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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
−11.94246149336716192017690448498, −10.66319002193887774722459892329, −9.736021550852362634362392378511, −8.951274740646641092472136573458, −8.010266538291002503177740215700, −6.55727876658949866684106947460, −5.59998821300806372191405409731, −4.18250541002276979321604479900, −3.16942872407691580346570537370, −0.849128789085722444070204926495,
0.849128789085722444070204926495, 3.16942872407691580346570537370, 4.18250541002276979321604479900, 5.59998821300806372191405409731, 6.55727876658949866684106947460, 8.010266538291002503177740215700, 8.951274740646641092472136573458, 9.736021550852362634362392378511, 10.66319002193887774722459892329, 11.94246149336716192017690448498
