| L(s) = 1 | + 7.42·2-s + 213.·3-s − 456.·4-s − 298.·5-s + 1.58e3·6-s + 7.72e3·7-s − 7.19e3·8-s + 2.58e4·9-s − 2.21e3·10-s + 9.28e3·11-s − 9.74e4·12-s − 1.22e5·13-s + 5.73e4·14-s − 6.36e4·15-s + 1.80e5·16-s + 1.92e5·18-s − 4.64e5·19-s + 1.36e5·20-s + 1.64e6·21-s + 6.89e4·22-s + 1.17e6·23-s − 1.53e6·24-s − 1.86e6·25-s − 9.10e5·26-s + 1.31e6·27-s − 3.52e6·28-s − 1.31e6·29-s + ⋯ |
| L(s) = 1 | + 0.328·2-s + 1.52·3-s − 0.892·4-s − 0.213·5-s + 0.499·6-s + 1.21·7-s − 0.621·8-s + 1.31·9-s − 0.0701·10-s + 0.191·11-s − 1.35·12-s − 1.18·13-s + 0.399·14-s − 0.324·15-s + 0.688·16-s + 0.431·18-s − 0.817·19-s + 0.190·20-s + 1.84·21-s + 0.0627·22-s + 0.874·23-s − 0.944·24-s − 0.954·25-s − 0.390·26-s + 0.476·27-s − 1.08·28-s − 0.345·29-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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 7.42T + 512T^{2} \) |
| 3 | \( 1 - 213.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 298.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.72e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 9.28e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.22e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 4.64e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.17e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.31e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.57e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.43e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.29e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.46e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.10e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.90e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.63e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.93e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.80e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.07e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.22e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.71e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.85e8T + 7.60e17T^{2} \) |
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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
−9.378093886057404616156866498167, −8.823479711925080890412975891265, −7.983025526632587432146125282001, −7.28881099488568142764885656835, −5.45505035728054889695270204643, −4.48598540677137769134751733905, −3.74321679045222017751757858723, −2.56615242763118377645418182293, −1.57560540472453819000937630971, 0,
1.57560540472453819000937630971, 2.56615242763118377645418182293, 3.74321679045222017751757858723, 4.48598540677137769134751733905, 5.45505035728054889695270204643, 7.28881099488568142764885656835, 7.983025526632587432146125282001, 8.823479711925080890412975891265, 9.378093886057404616156866498167
