| L(s) = 1 | + 2.22e3·3-s + 3.07e4·5-s − 3.41e5·7-s + 3.35e6·9-s + 9.18e6·11-s − 9.32e6·13-s + 6.82e7·15-s − 8.60e6·17-s − 2.27e8·19-s − 7.59e8·21-s + 5.58e8·23-s − 2.78e8·25-s + 3.91e9·27-s − 4.09e9·29-s + 2.31e8·31-s + 2.04e10·33-s − 1.04e10·35-s − 2.41e10·37-s − 2.07e10·39-s − 1.37e10·41-s + 1.90e9·43-s + 1.02e11·45-s − 2.27e10·47-s + 1.98e10·49-s − 1.91e10·51-s + 7.19e10·53-s + 2.82e11·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s + 0.878·5-s − 1.09·7-s + 2.10·9-s + 1.56·11-s − 0.535·13-s + 1.54·15-s − 0.0864·17-s − 1.10·19-s − 1.93·21-s + 0.786·23-s − 0.227·25-s + 1.94·27-s − 1.27·29-s + 0.0468·31-s + 2.75·33-s − 0.964·35-s − 1.54·37-s − 0.943·39-s − 0.453·41-s + 0.0460·43-s + 1.84·45-s − 0.307·47-s + 0.204·49-s − 0.152·51-s + 0.445·53-s + 1.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.057493019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.057493019\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.22e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 3.07e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.41e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 9.18e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 9.32e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 8.60e6T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.27e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.58e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.09e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 2.31e8T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.41e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 1.37e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.90e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 2.27e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 7.19e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 3.71e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 3.48e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 8.75e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.10e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 2.15e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.20e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.63e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.21e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 3.93e12T + 6.73e25T^{2} \) |
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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
−19.06146995850490333695964030123, −16.98880467838786691709757214886, −15.10450496132002133661116418687, −13.99239012874510052413366315131, −12.84418621935524834943199937424, −9.784037035096331188508543616076, −8.928015093795377593754458071171, −6.77693736343220153026919877280, −3.61491182189509783853227786530, −1.98847987996368312376160847260,
1.98847987996368312376160847260, 3.61491182189509783853227786530, 6.77693736343220153026919877280, 8.928015093795377593754458071171, 9.784037035096331188508543616076, 12.84418621935524834943199937424, 13.99239012874510052413366315131, 15.10450496132002133661116418687, 16.98880467838786691709757214886, 19.06146995850490333695964030123
