| L(s) = 1 | − 125.·3-s + 1.74e3·5-s + 2.40e3·7-s − 3.84e3·9-s + 2.08e4·11-s − 3.21e4·13-s − 2.19e5·15-s − 6.50e5·17-s + 9.41e5·19-s − 3.02e5·21-s + 5.22e5·23-s + 1.08e6·25-s + 2.96e6·27-s + 4.31e6·29-s + 4.82e6·31-s − 2.61e6·33-s + 4.18e6·35-s + 3.13e6·37-s + 4.04e6·39-s + 1.56e7·41-s − 2.35e7·43-s − 6.69e6·45-s + 5.41e7·47-s + 5.76e6·49-s + 8.18e7·51-s + 4.83e7·53-s + 3.62e7·55-s + ⋯ |
| L(s) = 1 | − 0.897·3-s + 1.24·5-s + 0.377·7-s − 0.195·9-s + 0.428·11-s − 0.311·13-s − 1.11·15-s − 1.88·17-s + 1.65·19-s − 0.339·21-s + 0.389·23-s + 0.554·25-s + 1.07·27-s + 1.13·29-s + 0.938·31-s − 0.384·33-s + 0.471·35-s + 0.274·37-s + 0.279·39-s + 0.866·41-s − 1.05·43-s − 0.243·45-s + 1.61·47-s + 0.142·49-s + 1.69·51-s + 0.841·53-s + 0.534·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.809261224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.809261224\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 + 125.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.74e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 2.08e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.21e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.50e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.41e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.22e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.31e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.13e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.35e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.41e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.83e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.54e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.75e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.35e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 7.20e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.35e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.65e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.28e6T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.51e9T + 7.60e17T^{2} \) |
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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
−13.44390760374663137408622189025, −12.00258537102367478109779520900, −11.09765872474578973212462931037, −9.916872828809593820466576351652, −8.765955014795698025127764386505, −6.85823193485481271837084724330, −5.81928383822772694006924475595, −4.75900480317033727744582683808, −2.47505681115703576202395916157, −0.920556206311076365743083171587,
0.920556206311076365743083171587, 2.47505681115703576202395916157, 4.75900480317033727744582683808, 5.81928383822772694006924475595, 6.85823193485481271837084724330, 8.765955014795698025127764386505, 9.916872828809593820466576351652, 11.09765872474578973212462931037, 12.00258537102367478109779520900, 13.44390760374663137408622189025
