| L(s) = 1 | + 3·3-s − 2·5-s + 7·7-s + 9·9-s − 8·11-s + 42·13-s − 6·15-s − 2·17-s − 124·19-s + 21·21-s − 76·23-s − 121·25-s + 27·27-s − 254·29-s + 72·31-s − 24·33-s − 14·35-s − 398·37-s + 126·39-s + 462·41-s + 212·43-s − 18·45-s + 264·47-s + 49·49-s − 6·51-s + 162·53-s + 16·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.178·5-s + 0.377·7-s + 1/3·9-s − 0.219·11-s + 0.896·13-s − 0.103·15-s − 0.0285·17-s − 1.49·19-s + 0.218·21-s − 0.689·23-s − 0.967·25-s + 0.192·27-s − 1.62·29-s + 0.417·31-s − 0.126·33-s − 0.0676·35-s − 1.76·37-s + 0.517·39-s + 1.75·41-s + 0.751·43-s − 0.0596·45-s + 0.819·47-s + 1/7·49-s − 0.0164·51-s + 0.419·53-s + 0.0392·55-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 76 T + p^{3} T^{2} \) |
| 29 | \( 1 + 254 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 462 T + p^{3} T^{2} \) |
| 43 | \( 1 - 212 T + p^{3} T^{2} \) |
| 47 | \( 1 - 264 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 772 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 - 236 T + p^{3} T^{2} \) |
| 73 | \( 1 - 418 T + p^{3} T^{2} \) |
| 79 | \( 1 + 552 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1190 T + p^{3} T^{2} \) |
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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
−8.808663546619771924353397181649, −8.039646923431951806614927307826, −7.43938534036221578896628142608, −6.32474896718453691904758307480, −5.56796356402839469481699290388, −4.27054325282074490283569675900, −3.76256429046002591026349162527, −2.44992868376034413475911252699, −1.56173991390130293832934536086, 0,
1.56173991390130293832934536086, 2.44992868376034413475911252699, 3.76256429046002591026349162527, 4.27054325282074490283569675900, 5.56796356402839469481699290388, 6.32474896718453691904758307480, 7.43938534036221578896628142608, 8.039646923431951806614927307826, 8.808663546619771924353397181649
