| L(s) = 1 | − 22.4·2-s + 161.·3-s − 6.03·4-s + 1.28e3·5-s − 3.62e3·6-s + 9.35e3·7-s + 1.16e4·8-s + 6.27e3·9-s − 2.88e4·10-s + 5.71e4·11-s − 972.·12-s − 1.47e5·13-s − 2.10e5·14-s + 2.06e5·15-s − 2.59e5·16-s − 1.41e5·18-s − 4.95e5·19-s − 7.75e3·20-s + 1.50e6·21-s − 1.28e6·22-s + 2.06e6·23-s + 1.87e6·24-s − 3.02e5·25-s + 3.31e6·26-s − 2.15e6·27-s − 5.64e4·28-s + 1.09e6·29-s + ⋯ |
| L(s) = 1 | − 0.994·2-s + 1.14·3-s − 0.0117·4-s + 0.919·5-s − 1.14·6-s + 1.47·7-s + 1.00·8-s + 0.318·9-s − 0.913·10-s + 1.17·11-s − 0.0135·12-s − 1.42·13-s − 1.46·14-s + 1.05·15-s − 0.988·16-s − 0.317·18-s − 0.872·19-s − 0.0108·20-s + 1.69·21-s − 1.16·22-s + 1.53·23-s + 1.15·24-s − 0.155·25-s + 1.42·26-s − 0.782·27-s − 0.0173·28-s + 0.287·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.966320923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.966320923\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 22.4T + 512T^{2} \) |
| 3 | \( 1 - 161.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.28e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.71e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.47e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 4.95e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.06e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.09e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.20e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.78e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.40e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.12e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.57e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.85e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.56e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.06e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.28e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.48e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.41e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.51e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.56e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.00e9T + 7.60e17T^{2} \) |
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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
−9.760282472067380224122205813168, −9.270162930321102544684059413390, −8.430419665493851039634394270648, −7.83640437399431402299455328511, −6.72260877749495974589849894895, −5.08657743003857330438196231949, −4.26167512140948523221429511691, −2.54700561186896304232142075223, −1.81500542598024096403739962880, −0.896178569189441583294613825409,
0.896178569189441583294613825409, 1.81500542598024096403739962880, 2.54700561186896304232142075223, 4.26167512140948523221429511691, 5.08657743003857330438196231949, 6.72260877749495974589849894895, 7.83640437399431402299455328511, 8.430419665493851039634394270648, 9.270162930321102544684059413390, 9.760282472067380224122205813168
