L(s) = 1 | + 32.0·2-s − 729·3-s − 7.16e3·4-s + 5.23e4·5-s − 2.33e4·6-s + 4.85e5·7-s − 4.91e5·8-s + 5.31e5·9-s + 1.67e6·10-s + 7.71e6·11-s + 5.22e6·12-s − 1.68e7·13-s + 1.55e7·14-s − 3.81e7·15-s + 4.29e7·16-s + 1.67e7·17-s + 1.70e7·18-s − 1.60e8·19-s − 3.75e8·20-s − 3.54e8·21-s + 2.47e8·22-s − 7.81e8·23-s + 3.58e8·24-s + 1.51e9·25-s − 5.39e8·26-s − 3.87e8·27-s − 3.48e9·28-s + ⋯ |
L(s) = 1 | + 0.353·2-s − 0.577·3-s − 0.874·4-s + 1.49·5-s − 0.204·6-s + 1.56·7-s − 0.663·8-s + 0.333·9-s + 0.530·10-s + 1.31·11-s + 0.505·12-s − 0.968·13-s + 0.552·14-s − 0.865·15-s + 0.639·16-s + 0.167·17-s + 0.117·18-s − 0.783·19-s − 1.31·20-s − 0.900·21-s + 0.464·22-s − 1.10·23-s + 0.383·24-s + 1.24·25-s − 0.342·26-s − 0.192·27-s − 1.36·28-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 | \( 1 - 32.0T + 8.19e3T^{2} \) |
| 5 | \( 1 - 5.23e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 4.85e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 7.71e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.68e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.67e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.60e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 7.81e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.88e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 5.16e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.18e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.96e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.22e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.14e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.15e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 1.57e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.15e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.50e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.68e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.28e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.33e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 6.70e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 4.23e12T + 6.73e25T^{2} \) |
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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
−9.714895755006858644465279579470, −9.118868623533122131636112546971, −7.891779536665510045589031618211, −6.44645213197205897821933452544, −5.52188021889494680954536621656, −4.88012258683499332352862392411, −3.90122938364644030803919484751, −2.00186954362490139581055029386, −1.44229553885386703697605895317, 0,
1.44229553885386703697605895317, 2.00186954362490139581055029386, 3.90122938364644030803919484751, 4.88012258683499332352862392411, 5.52188021889494680954536621656, 6.44645213197205897821933452544, 7.891779536665510045589031618211, 9.118868623533122131636112546971, 9.714895755006858644465279579470