| L(s) = 1 | − 5-s − 4·11-s + 2·13-s − 6·17-s − 19-s + 4·23-s + 25-s + 10·29-s + 4·31-s + 10·37-s − 6·41-s + 12·47-s − 7·49-s − 10·53-s + 4·55-s − 10·61-s − 2·65-s + 4·67-s + 10·73-s + 4·79-s − 12·83-s + 6·85-s − 14·89-s + 95-s − 2·97-s − 6·101-s − 16·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s + 1.85·29-s + 0.718·31-s + 1.64·37-s − 0.937·41-s + 1.75·47-s − 49-s − 1.37·53-s + 0.539·55-s − 1.28·61-s − 0.248·65-s + 0.488·67-s + 1.17·73-s + 0.450·79-s − 1.31·83-s + 0.650·85-s − 1.48·89-s + 0.102·95-s − 0.203·97-s − 0.597·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.81289355199641979551370165165, −6.75455879245676649787874977741, −6.43692522238243133727003650056, −5.40670078441511981591888331586, −4.65912254521612704782286522168, −4.16377166035292820952539672011, −2.95681836686443709979247590489, −2.53312602620037045170501575440, −1.19074048156047241483019349201, 0,
1.19074048156047241483019349201, 2.53312602620037045170501575440, 2.95681836686443709979247590489, 4.16377166035292820952539672011, 4.65912254521612704782286522168, 5.40670078441511981591888331586, 6.43692522238243133727003650056, 6.75455879245676649787874977741, 7.81289355199641979551370165165
