L(s) = 1 | − 3.15·2-s − 136.·3-s − 502.·4-s − 2.55e3·5-s + 430.·6-s − 9.39e3·7-s + 3.19e3·8-s − 1.04e3·9-s + 8.04e3·10-s − 4.40e4·11-s + 6.85e4·12-s + 2.96e4·14-s + 3.48e5·15-s + 2.46e5·16-s + 2.82e4·17-s + 3.28e3·18-s − 2.73e5·19-s + 1.28e6·20-s + 1.28e6·21-s + 1.38e5·22-s − 1.12e6·23-s − 4.36e5·24-s + 4.57e6·25-s + 2.82e6·27-s + 4.71e6·28-s − 1.63e6·29-s − 1.09e6·30-s + ⋯ |
L(s) = 1 | − 0.139·2-s − 0.973·3-s − 0.980·4-s − 1.82·5-s + 0.135·6-s − 1.47·7-s + 0.275·8-s − 0.0529·9-s + 0.254·10-s − 0.908·11-s + 0.954·12-s + 0.206·14-s + 1.77·15-s + 0.942·16-s + 0.0821·17-s + 0.00736·18-s − 0.482·19-s + 1.79·20-s + 1.44·21-s + 0.126·22-s − 0.841·23-s − 0.268·24-s + 2.34·25-s + 1.02·27-s + 1.45·28-s − 0.429·29-s − 0.247·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
good | 2 | \( 1 + 3.15T + 512T^{2} \) |
| 3 | \( 1 + 136.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.39e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.40e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 2.82e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.73e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.63e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.65e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.71e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 5.15e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.06e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.48e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.47e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.37e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.04e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.29e8T + 3.50e17T^{2} \) |
| 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
−10.70048741746803364187754854417, −9.659475294450197198394423233008, −8.444901642665896602182042141599, −7.59471045212571201007244253730, −6.33454831837771388742661077661, −5.14655760241795873421433752748, −4.05197769984591267103572959706, −3.16808608751511975633311226818, −0.49097619415568430083390199436, 0,
0.49097619415568430083390199436, 3.16808608751511975633311226818, 4.05197769984591267103572959706, 5.14655760241795873421433752748, 6.33454831837771388742661077661, 7.59471045212571201007244253730, 8.444901642665896602182042141599, 9.659475294450197198394423233008, 10.70048741746803364187754854417