| L(s) = 1 | + 0.0812·3-s + 2.29·5-s + 0.862·7-s − 2.99·9-s − 1.76·11-s − 4.62·13-s + 0.186·15-s − 7.59·17-s − 6.30·19-s + 0.0700·21-s − 6.92·23-s + 0.257·25-s − 0.486·27-s − 0.203·29-s + 7.39·31-s − 0.143·33-s + 1.97·35-s + 4.37·37-s − 0.375·39-s − 4.55·41-s + 6.44·43-s − 6.86·45-s + 12.7·47-s − 6.25·49-s − 0.616·51-s + 0.311·53-s − 4.03·55-s + ⋯ |
| L(s) = 1 | + 0.0468·3-s + 1.02·5-s + 0.326·7-s − 0.997·9-s − 0.530·11-s − 1.28·13-s + 0.0480·15-s − 1.84·17-s − 1.44·19-s + 0.0152·21-s − 1.44·23-s + 0.0514·25-s − 0.0936·27-s − 0.0378·29-s + 1.32·31-s − 0.0248·33-s + 0.334·35-s + 0.718·37-s − 0.0601·39-s − 0.711·41-s + 0.982·43-s − 1.02·45-s + 1.85·47-s − 0.893·49-s − 0.0863·51-s + 0.0428·53-s − 0.544·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0812T + 3T^{2} \) |
| 5 | \( 1 - 2.29T + 5T^{2} \) |
| 7 | \( 1 - 0.862T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 4.62T + 13T^{2} \) |
| 17 | \( 1 + 7.59T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 0.203T + 29T^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 + 4.55T + 41T^{2} \) |
| 43 | \( 1 - 6.44T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 0.311T + 53T^{2} \) |
| 59 | \( 1 - 3.24T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 - 4.17T + 73T^{2} \) |
| 79 | \( 1 + 0.537T + 79T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−9.191282857306792879151758570467, −8.485558857807025256005552484939, −7.71249947023680811199586709337, −6.50579971586448187438027339456, −5.99236190179356458761573646017, −4.99572680202039971413704078986, −4.21021122125824920019531440301, −2.37690816939709716191679771922, −2.30256538497453927538824944265, 0,
2.30256538497453927538824944265, 2.37690816939709716191679771922, 4.21021122125824920019531440301, 4.99572680202039971413704078986, 5.99236190179356458761573646017, 6.50579971586448187438027339456, 7.71249947023680811199586709337, 8.485558857807025256005552484939, 9.191282857306792879151758570467
