| L(s) = 1 | − 3·3-s + 10·5-s − 7·7-s + 9·9-s + 52·11-s + 10·13-s − 30·15-s − 54·17-s + 52·19-s + 21·21-s + 48·23-s − 25·25-s − 27·27-s + 186·29-s + 224·31-s − 156·33-s − 70·35-s − 94·37-s − 30·39-s − 478·41-s + 316·43-s + 90·45-s + 256·47-s + 49·49-s + 162·51-s + 66·53-s + 520·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.42·11-s + 0.213·13-s − 0.516·15-s − 0.770·17-s + 0.627·19-s + 0.218·21-s + 0.435·23-s − 1/5·25-s − 0.192·27-s + 1.19·29-s + 1.29·31-s − 0.822·33-s − 0.338·35-s − 0.417·37-s − 0.123·39-s − 1.82·41-s + 1.12·43-s + 0.298·45-s + 0.794·47-s + 1/7·49-s + 0.444·51-s + 0.171·53-s + 1.27·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.285390207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.285390207\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 94 T + p^{3} T^{2} \) |
| 41 | \( 1 + 478 T + p^{3} T^{2} \) |
| 43 | \( 1 - 316 T + p^{3} T^{2} \) |
| 47 | \( 1 - 256 T + p^{3} T^{2} \) |
| 53 | \( 1 - 66 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 342 T + p^{3} T^{2} \) |
| 67 | \( 1 + 668 T + p^{3} T^{2} \) |
| 71 | \( 1 + 272 T + p^{3} T^{2} \) |
| 73 | \( 1 + 86 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1360 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 366 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1554 T + p^{3} T^{2} \) |
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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
−9.251056264326039036322147908914, −8.721194493070375969892638030104, −7.41086749239516686365692461478, −6.40205310332690017886158206694, −6.24891294445059501707683358253, −5.08393444697847166847154228679, −4.21349463183984594806731434047, −3.07052182492375803578663171419, −1.79165915355476243365745597366, −0.817081095761583974296108904252,
0.817081095761583974296108904252, 1.79165915355476243365745597366, 3.07052182492375803578663171419, 4.21349463183984594806731434047, 5.08393444697847166847154228679, 6.24891294445059501707683358253, 6.40205310332690017886158206694, 7.41086749239516686365692461478, 8.721194493070375969892638030104, 9.251056264326039036322147908914
