L(s) = 1 | − 169.·2-s + 729·3-s + 2.05e4·4-s − 2.14e4·5-s − 1.23e5·6-s + 3.61e5·7-s − 2.09e6·8-s + 5.31e5·9-s + 3.63e6·10-s − 6.55e6·11-s + 1.49e7·12-s + 3.19e6·13-s − 6.12e7·14-s − 1.56e7·15-s + 1.86e8·16-s + 2.70e7·17-s − 9.00e7·18-s − 6.84e7·19-s − 4.40e8·20-s + 2.63e8·21-s + 1.11e9·22-s − 5.78e8·23-s − 1.52e9·24-s − 7.62e8·25-s − 5.41e8·26-s + 3.87e8·27-s + 7.42e9·28-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 0.577·3-s + 2.50·4-s − 0.612·5-s − 1.08·6-s + 1.16·7-s − 2.82·8-s + 0.333·9-s + 1.14·10-s − 1.11·11-s + 1.44·12-s + 0.183·13-s − 2.17·14-s − 0.353·15-s + 2.78·16-s + 0.271·17-s − 0.624·18-s − 0.333·19-s − 1.53·20-s + 0.669·21-s + 2.09·22-s − 0.814·23-s − 1.63·24-s − 0.624·25-s − 0.343·26-s + 0.192·27-s + 2.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 - 4.21e10T \) |
good | 2 | \( 1 + 169.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 2.14e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 3.61e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 6.55e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.19e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 2.70e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 6.84e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 5.78e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 3.44e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 9.42e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.03e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.19e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.12e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 3.14e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.87e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 5.01e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 5.06e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.47e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 5.72e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.61e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.15e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.78e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 3.76e11T + 6.73e25T^{2} \) |
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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
−9.824880728473388617280384860990, −8.522821244557925788960448305998, −8.058831836070132030359995308830, −7.56455680017789649546532057595, −6.22204597929045235691242169963, −4.59142618438619139425629483060, −2.96582648477584626514824797010, −2.04412654905221836736917116190, −1.07258993715244843308453709531, 0,
1.07258993715244843308453709531, 2.04412654905221836736917116190, 2.96582648477584626514824797010, 4.59142618438619139425629483060, 6.22204597929045235691242169963, 7.56455680017789649546532057595, 8.058831836070132030359995308830, 8.522821244557925788960448305998, 9.824880728473388617280384860990