| L(s) = 1 | + 1.83e4·3-s − 3.90e5·5-s − 1.60e7·7-s + 2.07e8·9-s − 7.88e8·11-s − 2.87e9·13-s − 7.16e9·15-s + 2.01e10·17-s + 2.28e10·19-s − 2.94e11·21-s + 6.65e11·23-s + 1.52e11·25-s + 1.43e12·27-s − 2.42e12·29-s − 8.75e12·31-s − 1.44e13·33-s + 6.28e12·35-s + 2.75e13·37-s − 5.27e13·39-s + 3.91e13·41-s + 1.29e14·43-s − 8.10e13·45-s + 3.00e14·47-s + 2.58e13·49-s + 3.69e14·51-s + 1.01e14·53-s + 3.07e14·55-s + ⋯ |
| L(s) = 1 | + 1.61·3-s − 0.447·5-s − 1.05·7-s + 1.60·9-s − 1.10·11-s − 0.977·13-s − 0.721·15-s + 0.700·17-s + 0.308·19-s − 1.70·21-s + 1.77·23-s + 0.200·25-s + 0.978·27-s − 0.898·29-s − 1.84·31-s − 1.78·33-s + 0.471·35-s + 1.28·37-s − 1.57·39-s + 0.766·41-s + 1.69·43-s − 0.718·45-s + 1.84·47-s + 0.111·49-s + 1.13·51-s + 0.224·53-s + 0.495·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.878928762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.878928762\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 3 | \( 1 - 1.83e4T + 1.29e8T^{2} \) |
| 7 | \( 1 + 1.60e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 7.88e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.87e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 2.01e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 2.28e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 6.65e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.42e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 8.75e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.75e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 3.91e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.29e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 3.00e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 1.01e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 9.67e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.14e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.04e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 7.35e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 6.99e14T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.08e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.23e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.21e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 9.41e16T + 5.95e33T^{2} \) |
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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
−10.87051837267784035810637393496, −9.628072429717692876876042779351, −9.068156693209783012256709418281, −7.67562512229547918083057229439, −7.27594229690218473955550249169, −5.41509828113008111056014051850, −3.92676362674878143663097501561, −2.99614205837655954279247553467, −2.37130225801542477972107057462, −0.67277366519210393357403361856,
0.67277366519210393357403361856, 2.37130225801542477972107057462, 2.99614205837655954279247553467, 3.92676362674878143663097501561, 5.41509828113008111056014051850, 7.27594229690218473955550249169, 7.67562512229547918083057229439, 9.068156693209783012256709418281, 9.628072429717692876876042779351, 10.87051837267784035810637393496
