L(s) = 1 | + 0.396·2-s + 27·3-s − 127.·4-s + 247.·5-s + 10.7·6-s − 652.·7-s − 101.·8-s + 729·9-s + 98.1·10-s − 2.47e3·11-s − 3.45e3·12-s + 9.41e3·13-s − 259.·14-s + 6.67e3·15-s + 1.63e4·16-s + 1.25e4·17-s + 289.·18-s − 4.13e4·19-s − 3.16e4·20-s − 1.76e4·21-s − 983.·22-s − 5.97e4·23-s − 2.74e3·24-s − 1.70e4·25-s + 3.73e3·26-s + 1.96e4·27-s + 8.34e4·28-s + ⋯ |
L(s) = 1 | + 0.0350·2-s + 0.577·3-s − 0.998·4-s + 0.884·5-s + 0.0202·6-s − 0.719·7-s − 0.0701·8-s + 0.333·9-s + 0.0310·10-s − 0.561·11-s − 0.576·12-s + 1.18·13-s − 0.0252·14-s + 0.510·15-s + 0.996·16-s + 0.620·17-s + 0.0116·18-s − 1.38·19-s − 0.883·20-s − 0.415·21-s − 0.0196·22-s − 1.02·23-s − 0.0404·24-s − 0.217·25-s + 0.0417·26-s + 0.192·27-s + 0.718·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
good | 2 | \( 1 - 0.396T + 128T^{2} \) |
| 5 | \( 1 - 247.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 652.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.41e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.25e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.13e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.12e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.79e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.47e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.40e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.13e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.10e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 8.10e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.83e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.35e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.61e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.37e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.23e6T + 8.07e13T^{2} \) |
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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
−10.35135429529233991858307727334, −9.977953598197719325186624675680, −8.790686303821147788760426793571, −8.208899116856546786223613735871, −6.51650243417015005905868689061, −5.57161710558010497553631110556, −4.18423462221505586369392273888, −3.07934233579018698658220010933, −1.58847622864308660485683554159, 0,
1.58847622864308660485683554159, 3.07934233579018698658220010933, 4.18423462221505586369392273888, 5.57161710558010497553631110556, 6.51650243417015005905868689061, 8.208899116856546786223613735871, 8.790686303821147788760426793571, 9.977953598197719325186624675680, 10.35135429529233991858307727334