| L(s) = 1 | − 17.1·2-s + 137.·3-s − 217.·4-s − 1.24e3·5-s − 2.36e3·6-s − 1.05e4·7-s + 1.25e4·8-s − 717.·9-s + 2.14e4·10-s − 4.94e4·11-s − 3.00e4·12-s + 5.43e4·13-s + 1.80e5·14-s − 1.71e5·15-s − 1.03e5·16-s + 1.23e4·18-s + 7.16e5·19-s + 2.71e5·20-s − 1.45e6·21-s + 8.47e5·22-s + 2.03e5·23-s + 1.72e6·24-s − 3.95e5·25-s − 9.32e5·26-s − 2.80e6·27-s + 2.29e6·28-s + 6.48e6·29-s + ⋯ |
| L(s) = 1 | − 0.757·2-s + 0.981·3-s − 0.425·4-s − 0.892·5-s − 0.743·6-s − 1.66·7-s + 1.08·8-s − 0.0364·9-s + 0.676·10-s − 1.01·11-s − 0.417·12-s + 0.527·13-s + 1.25·14-s − 0.876·15-s − 0.393·16-s + 0.0276·18-s + 1.26·19-s + 0.379·20-s − 1.63·21-s + 0.771·22-s + 0.151·23-s + 1.06·24-s − 0.202·25-s − 0.400·26-s − 1.01·27-s + 0.706·28-s + 1.70·29-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 | \( 1 + 17.1T + 512T^{2} \) |
| 3 | \( 1 - 137.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.24e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.94e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.43e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.16e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.03e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.48e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.52e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.86e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.92e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.47e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 7.91e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.83e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.19e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.69e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.75e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.86e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.69e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.25e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.13e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.81e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.61e9T + 7.60e17T^{2} \) |
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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
−9.638585055939812539951799080847, −8.786224965335946944103792851646, −8.014915425816065634780598190615, −7.35869691836329433952856899909, −5.96850234112929839909321807167, −4.45657673748000959804956860844, −3.35474590951825442769300954877, −2.74226462487546993631445601057, −0.881355223333240276877649861867, 0,
0.881355223333240276877649861867, 2.74226462487546993631445601057, 3.35474590951825442769300954877, 4.45657673748000959804956860844, 5.96850234112929839909321807167, 7.35869691836329433952856899909, 8.014915425816065634780598190615, 8.786224965335946944103792851646, 9.638585055939812539951799080847
