| L(s) = 1 | + 0.732·3-s − 2.73·5-s − 2.73·7-s − 2.46·9-s − 3·11-s − 3·13-s − 2·15-s + 6.46·17-s − 0.535·19-s − 2·21-s − 23-s + 2.46·25-s − 4·27-s + 6.19·29-s − 5·31-s − 2.19·33-s + 7.46·35-s − 1.46·37-s − 2.19·39-s + 9.92·41-s − 43-s + 6.73·45-s − 10.3·47-s + 0.464·49-s + 4.73·51-s − 2.46·53-s + 8.19·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 1.22·5-s − 1.03·7-s − 0.821·9-s − 0.904·11-s − 0.832·13-s − 0.516·15-s + 1.56·17-s − 0.122·19-s − 0.436·21-s − 0.208·23-s + 0.492·25-s − 0.769·27-s + 1.15·29-s − 0.898·31-s − 0.382·33-s + 1.26·35-s − 0.240·37-s − 0.351·39-s + 1.55·41-s − 0.152·43-s + 1.00·45-s − 1.51·47-s + 0.0663·49-s + 0.662·51-s − 0.338·53-s + 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 2.46T + 53T^{2} \) |
| 59 | \( 1 + 8.92T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 7.92T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 0.196T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−11.08530910709519606745948714197, −10.07978847120298318421933090561, −9.194640696382686825391024656413, −7.957388207868387939513549056761, −7.62374494576678547469668587769, −6.21353203156467243317252083850, −5.00399554891788388508458114283, −3.55112505413581355711905562797, −2.82403314931639398014294180141, 0,
2.82403314931639398014294180141, 3.55112505413581355711905562797, 5.00399554891788388508458114283, 6.21353203156467243317252083850, 7.62374494576678547469668587769, 7.957388207868387939513549056761, 9.194640696382686825391024656413, 10.07978847120298318421933090561, 11.08530910709519606745948714197
