| L(s) = 1 | − 16·5-s + 7·7-s − 8·11-s + 28·13-s − 54·17-s + 110·19-s + 48·23-s + 131·25-s + 110·29-s − 12·31-s − 112·35-s − 246·37-s − 182·41-s − 128·43-s + 324·47-s + 49·49-s + 162·53-s + 128·55-s + 810·59-s − 488·61-s − 448·65-s − 244·67-s − 768·71-s − 702·73-s − 56·77-s − 440·79-s − 1.30e3·83-s + ⋯ |
| L(s) = 1 | − 1.43·5-s + 0.377·7-s − 0.219·11-s + 0.597·13-s − 0.770·17-s + 1.32·19-s + 0.435·23-s + 1.04·25-s + 0.704·29-s − 0.0695·31-s − 0.540·35-s − 1.09·37-s − 0.693·41-s − 0.453·43-s + 1.00·47-s + 1/7·49-s + 0.419·53-s + 0.313·55-s + 1.78·59-s − 1.02·61-s − 0.854·65-s − 0.444·67-s − 1.28·71-s − 1.12·73-s − 0.0828·77-s − 0.626·79-s − 1.72·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 110 T + p^{3} T^{2} \) |
| 31 | \( 1 + 12 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 182 T + p^{3} T^{2} \) |
| 43 | \( 1 + 128 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 810 T + p^{3} T^{2} \) |
| 61 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 702 T + p^{3} T^{2} \) |
| 79 | \( 1 + 440 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 + 730 T + p^{3} T^{2} \) |
| 97 | \( 1 - 294 T + p^{3} T^{2} \) |
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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
−8.869566010380685179149485499063, −8.395537087182224188727401870480, −7.46237582387209133272813529406, −6.90977955704289109978761614351, −5.59589080947427806353041911109, −4.64731813094226005976160895793, −3.80631318277586038798836781027, −2.88076824465994371591170499655, −1.26898321762764660535375187153, 0,
1.26898321762764660535375187153, 2.88076824465994371591170499655, 3.80631318277586038798836781027, 4.64731813094226005976160895793, 5.59589080947427806353041911109, 6.90977955704289109978761614351, 7.46237582387209133272813529406, 8.395537087182224188727401870480, 8.869566010380685179149485499063
