| L(s) = 1 | − 42.0·2-s − 256.·3-s + 1.25e3·4-s + 1.40e3·5-s + 1.08e4·6-s − 5.13e3·7-s − 3.13e4·8-s + 4.62e4·9-s − 5.91e4·10-s + 2.65e4·11-s − 3.22e5·12-s + 7.14e4·13-s + 2.16e5·14-s − 3.61e5·15-s + 6.74e5·16-s − 8.35e4·17-s − 1.94e6·18-s − 5.48e5·19-s + 1.76e6·20-s + 1.31e6·21-s − 1.11e6·22-s + 1.15e6·23-s + 8.04e6·24-s + 2.72e4·25-s − 3.00e6·26-s − 6.82e6·27-s − 6.46e6·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.83·3-s + 2.45·4-s + 1.00·5-s + 3.40·6-s − 0.808·7-s − 2.70·8-s + 2.34·9-s − 1.87·10-s + 0.546·11-s − 4.49·12-s + 0.693·13-s + 1.50·14-s − 1.84·15-s + 2.57·16-s − 0.242·17-s − 4.36·18-s − 0.965·19-s + 2.47·20-s + 1.48·21-s − 1.01·22-s + 0.864·23-s + 4.95·24-s + 0.0139·25-s − 1.28·26-s − 2.47·27-s − 1.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8.35e4T \) |
| good | 2 | \( 1 + 42.0T + 512T^{2} \) |
| 3 | \( 1 + 256.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.40e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.13e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.14e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.48e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.15e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.05e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.50e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.75e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.15e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.02e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.95e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.34e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.45e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.71e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.75e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.33e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.87e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.41e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.74617914817934616029540958849, −15.74625118886963632112456694456, −12.72126114348821388618046082561, −11.23240595320376213511309813856, −10.34105374452628142326963725185, −9.231449938266936398061284756306, −6.80615706385180649153702521673, −6.01247830173674070548198063867, −1.46726334419730167440098000473, 0,
1.46726334419730167440098000473, 6.01247830173674070548198063867, 6.80615706385180649153702521673, 9.231449938266936398061284756306, 10.34105374452628142326963725185, 11.23240595320376213511309813856, 12.72126114348821388618046082561, 15.74625118886963632112456694456, 16.74617914817934616029540958849
