| L(s) = 1 | − 5-s − 2·7-s + 6·11-s + 4·13-s + 6·17-s − 4·19-s + 8·23-s + 25-s + 2·29-s + 31-s + 2·35-s + 8·37-s − 8·41-s − 3·49-s + 6·53-s − 6·55-s − 6·59-s − 10·61-s − 4·65-s + 8·67-s + 10·73-s − 12·77-s + 4·83-s − 6·85-s − 8·91-s + 4·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.80·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.179·31-s + 0.338·35-s + 1.31·37-s − 1.24·41-s − 3/7·49-s + 0.824·53-s − 0.809·55-s − 0.781·59-s − 1.28·61-s − 0.496·65-s + 0.977·67-s + 1.17·73-s − 1.36·77-s + 0.439·83-s − 0.650·85-s − 0.838·91-s + 0.410·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.263160607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.263160607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.89255783106353, −13.31817609687713, −12.89676063791215, −12.34104650022038, −11.96879125134793, −11.38296105105809, −11.04251733667256, −10.38924586378829, −9.882673867754673, −9.242467632460937, −8.982457376323677, −8.402239514068716, −7.876049029940205, −7.170888399617107, −6.646345071804026, −6.291471318193739, −5.853783008249889, −4.967909744495364, −4.445024029016205, −3.695877667278979, −3.450632882571195, −2.908642724057448, −1.849252100426376, −1.108315091209293, −0.6970852214645254,
0.6970852214645254, 1.108315091209293, 1.849252100426376, 2.908642724057448, 3.450632882571195, 3.695877667278979, 4.445024029016205, 4.967909744495364, 5.853783008249889, 6.291471318193739, 6.646345071804026, 7.170888399617107, 7.876049029940205, 8.402239514068716, 8.982457376323677, 9.242467632460937, 9.882673867754673, 10.38924586378829, 11.04251733667256, 11.38296105105809, 11.96879125134793, 12.34104650022038, 12.89676063791215, 13.31817609687713, 13.89255783106353
