| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s − 4·13-s − 15-s + 7·17-s − 9·23-s − 4·25-s + 27-s − 10·29-s − 8·31-s − 2·33-s − 7·37-s − 4·39-s − 12·41-s + 43-s − 45-s + 8·47-s + 7·51-s − 6·53-s + 2·55-s − 61-s + 4·65-s + 12·67-s − 9·69-s + 3·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.258·15-s + 1.69·17-s − 1.87·23-s − 4/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.348·33-s − 1.15·37-s − 0.640·39-s − 1.87·41-s + 0.152·43-s − 0.149·45-s + 1.16·47-s + 0.980·51-s − 0.824·53-s + 0.269·55-s − 0.128·61-s + 0.496·65-s + 1.46·67-s − 1.08·69-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| 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
−13.95901946475224, −13.45897738237540, −12.78895840312504, −12.45196394770521, −12.01402127409271, −11.63391431542537, −10.95888884048018, −10.36928950888676, −9.908554733409675, −9.687590585093643, −9.060509501481867, −8.403619511156979, −7.927549125226278, −7.528007846241551, −7.338098724356850, −6.582005194160735, −5.707549658914499, −5.463868607132126, −4.968583923139767, −4.060049974992969, −3.674781372647557, −3.338707222725625, −2.435570174903321, −1.978388898915852, −1.402926377536306, 0, 0,
1.402926377536306, 1.978388898915852, 2.435570174903321, 3.338707222725625, 3.674781372647557, 4.060049974992969, 4.968583923139767, 5.463868607132126, 5.707549658914499, 6.582005194160735, 7.338098724356850, 7.528007846241551, 7.927549125226278, 8.403619511156979, 9.060509501481867, 9.687590585093643, 9.908554733409675, 10.36928950888676, 10.95888884048018, 11.63391431542537, 12.01402127409271, 12.45196394770521, 12.78895840312504, 13.45897738237540, 13.95901946475224
