L(s) = 1 | − 5-s − 11-s − 2·13-s − 6·17-s − 4·19-s − 8·23-s + 25-s + 6·29-s + 4·31-s + 2·37-s − 6·41-s − 12·43-s − 2·53-s + 55-s − 4·59-s + 2·61-s + 2·65-s + 8·67-s − 8·71-s − 2·73-s − 12·79-s + 8·83-s + 6·85-s + 18·89-s + 4·95-s + 10·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.82·43-s − 0.274·53-s + 0.134·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s + 0.977·67-s − 0.949·71-s − 0.234·73-s − 1.35·79-s + 0.878·83-s + 0.650·85-s + 1.90·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−12.81102363153329, −12.13058754829754, −11.63829427213522, −11.61256635131094, −10.83199994062486, −10.34214953867698, −10.11136719159429, −9.638432829024869, −8.875703860479089, −8.577299785683106, −8.186740892174980, −7.756693573787471, −7.139093754438695, −6.681886545510804, −6.255492170324994, −5.909698219530123, −4.989886419128668, −4.644392919255237, −4.418984880690482, −3.629428539635306, −3.249944857527450, −2.371289950391355, −2.203511510497415, −1.496101833724509, −0.5164226919356775, 0,
0.5164226919356775, 1.496101833724509, 2.203511510497415, 2.371289950391355, 3.249944857527450, 3.629428539635306, 4.418984880690482, 4.644392919255237, 4.989886419128668, 5.909698219530123, 6.255492170324994, 6.681886545510804, 7.139093754438695, 7.756693573787471, 8.186740892174980, 8.577299785683106, 8.875703860479089, 9.638432829024869, 10.11136719159429, 10.34214953867698, 10.83199994062486, 11.61256635131094, 11.63829427213522, 12.13058754829754, 12.81102363153329