| L(s) = 1 | − 39.2·2-s − 12.8·3-s + 1.02e3·4-s − 2.41e3·5-s + 503.·6-s − 2.30e3·7-s − 2.02e4·8-s − 1.95e4·9-s + 9.47e4·10-s − 8.14e4·11-s − 1.31e4·12-s − 5.93e4·13-s + 9.03e4·14-s + 3.09e4·15-s + 2.68e5·16-s + 7.65e5·18-s − 3.66e5·19-s − 2.48e6·20-s + 2.95e4·21-s + 3.19e6·22-s − 1.24e6·23-s + 2.60e5·24-s + 3.87e6·25-s + 2.33e6·26-s + 5.03e5·27-s − 2.36e6·28-s − 1.35e6·29-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.0915·3-s + 2.00·4-s − 1.72·5-s + 0.158·6-s − 0.362·7-s − 1.74·8-s − 0.991·9-s + 2.99·10-s − 1.67·11-s − 0.183·12-s − 0.576·13-s + 0.628·14-s + 0.158·15-s + 1.02·16-s + 1.71·18-s − 0.644·19-s − 3.46·20-s + 0.0331·21-s + 2.90·22-s − 0.930·23-s + 0.159·24-s + 1.98·25-s + 1.00·26-s + 0.182·27-s − 0.727·28-s − 0.356·29-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 | \( 1 + 39.2T + 512T^{2} \) |
| 3 | \( 1 + 12.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.41e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.30e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.14e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.93e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 3.66e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.35e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.78e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.36e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.02e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.33e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.90e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.82e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.09e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.68e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.42e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.50e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.12e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.17e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.46e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.57e8T + 7.60e17T^{2} \) |
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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
−9.750634651420331977698968949090, −8.489293667543190969031239224043, −8.039139601696732519659407104141, −7.44290792375340459852102657603, −6.25655594948154917810583685847, −4.74489071828924281455497465904, −3.20308484720452824719061205525, −2.32501724964070354895021983548, −0.51525299187980632165173549955, 0,
0.51525299187980632165173549955, 2.32501724964070354895021983548, 3.20308484720452824719061205525, 4.74489071828924281455497465904, 6.25655594948154917810583685847, 7.44290792375340459852102657603, 8.039139601696732519659407104141, 8.489293667543190969031239224043, 9.750634651420331977698968949090
