| L(s) = 1 | − 241.·3-s + 625·5-s − 4.25e3·7-s + 3.85e4·9-s + 2.16e4·11-s + 1.19e5·13-s − 1.50e5·15-s + 4.40e5·17-s + 2.28e5·19-s + 1.02e6·21-s + 2.48e5·23-s + 3.90e5·25-s − 4.55e6·27-s + 5.78e6·29-s + 5.73e6·31-s − 5.23e6·33-s − 2.65e6·35-s + 1.78e7·37-s − 2.89e7·39-s − 2.04e7·41-s + 3.39e7·43-s + 2.41e7·45-s − 3.52e7·47-s − 2.22e7·49-s − 1.06e8·51-s − 1.81e6·53-s + 1.35e7·55-s + ⋯ |
| L(s) = 1 | − 1.72·3-s + 0.447·5-s − 0.669·7-s + 1.95·9-s + 0.446·11-s + 1.16·13-s − 0.769·15-s + 1.27·17-s + 0.402·19-s + 1.15·21-s + 0.185·23-s + 0.200·25-s − 1.64·27-s + 1.51·29-s + 1.11·31-s − 0.767·33-s − 0.299·35-s + 1.56·37-s − 2.00·39-s − 1.13·41-s + 1.51·43-s + 0.876·45-s − 1.05·47-s − 0.552·49-s − 2.20·51-s − 0.0316·53-s + 0.199·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.632117587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632117587\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| good | 3 | \( 1 + 241.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 4.25e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.16e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.19e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.40e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.48e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.78e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.73e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.78e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.39e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.52e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.81e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.39e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.15e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.49e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.16e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.90e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.59e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.20e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.31e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.56e8T + 7.60e17T^{2} \) |
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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
−10.16469692897461809952149833128, −9.539323912638222716566554026135, −8.104944248900160389485590315231, −6.70507103966074733944024041008, −6.22439989481092849487820555866, −5.44457439448456880129963968573, −4.38064181300144474518611605434, −3.11246164438612864442876456994, −1.27899035706155543972102686493, −0.72701248944504607499860558144,
0.72701248944504607499860558144, 1.27899035706155543972102686493, 3.11246164438612864442876456994, 4.38064181300144474518611605434, 5.44457439448456880129963968573, 6.22439989481092849487820555866, 6.70507103966074733944024041008, 8.104944248900160389485590315231, 9.539323912638222716566554026135, 10.16469692897461809952149833128
