L(s) = 1 | − 9.57·2-s − 3.76·3-s − 420.·4-s + 1.87e3·5-s + 36.0·6-s − 1.55e3·7-s + 8.92e3·8-s − 1.96e4·9-s − 1.79e4·10-s + 6.88e4·11-s + 1.58e3·12-s − 5.31e4·13-s + 1.48e4·14-s − 7.04e3·15-s + 1.29e5·16-s + 1.88e5·18-s + 6.43e4·19-s − 7.87e5·20-s + 5.84e3·21-s − 6.59e5·22-s − 2.42e6·23-s − 3.35e4·24-s + 1.55e6·25-s + 5.08e5·26-s + 1.48e5·27-s + 6.52e5·28-s + 4.67e6·29-s + ⋯ |
L(s) = 1 | − 0.423·2-s − 0.0268·3-s − 0.820·4-s + 1.34·5-s + 0.0113·6-s − 0.244·7-s + 0.770·8-s − 0.999·9-s − 0.567·10-s + 1.41·11-s + 0.0220·12-s − 0.515·13-s + 0.103·14-s − 0.0359·15-s + 0.494·16-s + 0.422·18-s + 0.113·19-s − 1.10·20-s + 0.00655·21-s − 0.599·22-s − 1.81·23-s − 0.0206·24-s + 0.796·25-s + 0.218·26-s + 0.0536·27-s + 0.200·28-s + 1.22·29-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 | \( 1 + 9.57T + 512T^{2} \) |
| 3 | \( 1 + 3.76T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.87e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.55e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.88e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.31e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 6.43e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.15e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.76e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.09e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.81e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.87e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.47e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.26e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.41e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.37e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.06e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.45e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.94e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.15e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.84e8T + 7.60e17T^{2} \) |
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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
−9.796888315659731718037771764050, −8.917437455102680232085859107100, −8.208207591679858465611125509544, −6.65482816480763596597167465157, −5.87928793203451150395495970071, −4.87522103093538852341419163929, −3.63712408280707202669866851682, −2.24428703805061349956094750496, −1.19258587369311411653045247679, 0,
1.19258587369311411653045247679, 2.24428703805061349956094750496, 3.63712408280707202669866851682, 4.87522103093538852341419163929, 5.87928793203451150395495970071, 6.65482816480763596597167465157, 8.208207591679858465611125509544, 8.917437455102680232085859107100, 9.796888315659731718037771764050