| L(s) = 1 | − 4.11·2-s − 81·3-s − 495.·4-s − 2.18e3·5-s + 333.·6-s − 4.53e3·7-s + 4.14e3·8-s + 6.56e3·9-s + 8.99e3·10-s − 2.85e4·11-s + 4.00e4·12-s − 1.02e5·13-s + 1.86e4·14-s + 1.77e5·15-s + 2.36e5·16-s + 1.21e5·17-s − 2.70e4·18-s − 2.12e5·19-s + 1.08e6·20-s + 3.67e5·21-s + 1.17e5·22-s − 1.13e6·23-s − 3.35e5·24-s + 2.82e6·25-s + 4.21e5·26-s − 5.31e5·27-s + 2.24e6·28-s + ⋯ |
| L(s) = 1 | − 0.181·2-s − 0.577·3-s − 0.966·4-s − 1.56·5-s + 0.105·6-s − 0.714·7-s + 0.357·8-s + 0.333·9-s + 0.284·10-s − 0.587·11-s + 0.558·12-s − 0.994·13-s + 0.129·14-s + 0.902·15-s + 0.901·16-s + 0.351·17-s − 0.0606·18-s − 0.374·19-s + 1.51·20-s + 0.412·21-s + 0.106·22-s − 0.844·23-s − 0.206·24-s + 1.44·25-s + 0.180·26-s − 0.192·27-s + 0.690·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 + 4.11T + 512T^{2} \) |
| 5 | \( 1 + 2.18e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.53e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.85e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.02e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.21e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.12e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.13e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.92e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.24e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.14e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.27e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.07e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.37e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 2.27e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.95e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.08e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.10e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.80e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.16e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.17e8T + 7.60e17T^{2} \) |
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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
−10.37772530660236117994833283523, −9.675771080555322843123977913409, −8.254067185694345475140895895260, −7.69431738207651468321279428009, −6.37455297174854217697576300204, −4.88487632568956218089667422375, −4.21044971478722605525978939423, −2.97619431058621553567212803575, −0.72477139571564419765602549789, 0,
0.72477139571564419765602549789, 2.97619431058621553567212803575, 4.21044971478722605525978939423, 4.88487632568956218089667422375, 6.37455297174854217697576300204, 7.69431738207651468321279428009, 8.254067185694345475140895895260, 9.675771080555322843123977913409, 10.37772530660236117994833283523
