| L(s) = 1 | − 3·3-s − 5·5-s − 16·7-s + 9·9-s + 24·11-s + 14·13-s + 15·15-s − 18·17-s + 36·19-s + 48·21-s − 104·23-s + 25·25-s − 27·27-s + 250·29-s + 28·31-s − 72·33-s + 80·35-s + 54·37-s − 42·39-s + 354·41-s + 228·43-s − 45·45-s − 408·47-s − 87·49-s + 54·51-s − 262·53-s − 120·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.863·7-s + 1/3·9-s + 0.657·11-s + 0.298·13-s + 0.258·15-s − 0.256·17-s + 0.434·19-s + 0.498·21-s − 0.942·23-s + 1/5·25-s − 0.192·27-s + 1.60·29-s + 0.162·31-s − 0.379·33-s + 0.386·35-s + 0.239·37-s − 0.172·39-s + 1.34·41-s + 0.808·43-s − 0.149·45-s − 1.26·47-s − 0.253·49-s + 0.148·51-s − 0.679·53-s − 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 104 T + p^{3} T^{2} \) |
| 29 | \( 1 - 250 T + p^{3} T^{2} \) |
| 31 | \( 1 - 28 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 - 354 T + p^{3} T^{2} \) |
| 43 | \( 1 - 228 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 262 T + p^{3} T^{2} \) |
| 59 | \( 1 + 64 T + p^{3} T^{2} \) |
| 61 | \( 1 + 374 T + p^{3} T^{2} \) |
| 67 | \( 1 - 300 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1016 T + p^{3} T^{2} \) |
| 73 | \( 1 - 274 T + p^{3} T^{2} \) |
| 79 | \( 1 + 788 T + p^{3} T^{2} \) |
| 83 | \( 1 + 396 T + p^{3} T^{2} \) |
| 89 | \( 1 - 786 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1086 T + p^{3} T^{2} \) |
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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
−9.364137416451308212340495018574, −8.387918744127479996564660815500, −7.44654875128450456121083910392, −6.48405530058010351210656023939, −6.01313784909254016315208688000, −4.71576113699755209857606180820, −3.88819984660222698221772685721, −2.82706634280464347355548352150, −1.21435010977202980235033478309, 0,
1.21435010977202980235033478309, 2.82706634280464347355548352150, 3.88819984660222698221772685721, 4.71576113699755209857606180820, 6.01313784909254016315208688000, 6.48405530058010351210656023939, 7.44654875128450456121083910392, 8.387918744127479996564660815500, 9.364137416451308212340495018574
