| 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
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 22.4T + 512T^{2} \) |
| 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
−10.01016649542275435802107367182, −8.892240579888060816270532927520, −7.76001239807326516269980081102, −7.09737659161088507925549522805, −5.89406639033647760509667709420, −4.86830609954576653913118353969, −3.72192336977984732281818714172, −2.23327254319631461557676415891, −0.29950462214626565905603532415, 0,
0.29950462214626565905603532415, 2.23327254319631461557676415891, 3.72192336977984732281818714172, 4.86830609954576653913118353969, 5.89406639033647760509667709420, 7.09737659161088507925549522805, 7.76001239807326516269980081102, 8.892240579888060816270532927520, 10.01016649542275435802107367182
