| L(s) = 1 | + 5-s + 7-s − 2·11-s − 2·13-s + 2·17-s − 8·19-s − 2·23-s + 25-s − 8·29-s − 8·31-s + 35-s − 6·37-s + 6·41-s + 4·43-s + 4·47-s + 49-s − 2·55-s − 4·59-s − 2·61-s − 2·65-s − 4·67-s + 6·71-s − 6·73-s − 2·77-s − 16·83-s + 2·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.417·23-s + 1/5·25-s − 1.48·29-s − 1.43·31-s + 0.169·35-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.269·55-s − 0.520·59-s − 0.256·61-s − 0.248·65-s − 0.488·67-s + 0.712·71-s − 0.702·73-s − 0.227·77-s − 1.75·83-s + 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−8.600446367510877122148794610191, −7.68516014416835048250854842069, −7.14981956534707632378469862500, −6.03312946153378432457636135165, −5.50026028907130812605313341968, −4.57625495297883443918276227368, −3.70932754536319705699105478693, −2.46766077498887876942283623243, −1.75193199532647892290011836792, 0,
1.75193199532647892290011836792, 2.46766077498887876942283623243, 3.70932754536319705699105478693, 4.57625495297883443918276227368, 5.50026028907130812605313341968, 6.03312946153378432457636135165, 7.14981956534707632378469862500, 7.68516014416835048250854842069, 8.600446367510877122148794610191
