L(s) = 1 | − 44.6·2-s − 81·3-s + 1.48e3·4-s − 714.·5-s + 3.61e3·6-s − 1.00e4·7-s − 4.33e4·8-s + 6.56e3·9-s + 3.19e4·10-s − 1.69e4·11-s − 1.20e5·12-s − 1.93e4·13-s + 4.50e5·14-s + 5.78e4·15-s + 1.17e6·16-s + 3.36e5·17-s − 2.93e5·18-s − 7.88e5·19-s − 1.05e6·20-s + 8.16e5·21-s + 7.55e5·22-s + 1.24e5·23-s + 3.51e6·24-s − 1.44e6·25-s + 8.62e5·26-s − 5.31e5·27-s − 1.49e7·28-s + ⋯ |
L(s) = 1 | − 1.97·2-s − 0.577·3-s + 2.89·4-s − 0.511·5-s + 1.13·6-s − 1.58·7-s − 3.74·8-s + 0.333·9-s + 1.00·10-s − 0.348·11-s − 1.67·12-s − 0.187·13-s + 3.13·14-s + 0.295·15-s + 4.48·16-s + 0.978·17-s − 0.657·18-s − 1.38·19-s − 1.47·20-s + 0.916·21-s + 0.687·22-s + 0.0924·23-s + 2.16·24-s − 0.738·25-s + 0.370·26-s − 0.192·27-s − 4.59·28-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 | \( 1 + 44.6T + 512T^{2} \) |
| 5 | \( 1 + 714.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.00e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.69e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.36e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.24e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.52e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.79e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.64e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.96e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.00e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.32e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.36e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 4.31e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.79e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.26e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.95e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.38e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.18e9T + 7.60e17T^{2} \) |
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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
−10.22093558811263576365367135105, −9.684759376246685710824238226498, −8.580952454804470183538270872880, −7.55224580497059979775187575187, −6.67087762072626538161918478009, −5.87244582432605089826166923091, −3.51322416617707031377994988097, −2.31349123789614063390489146856, −0.75268128211626770271952768569, 0,
0.75268128211626770271952768569, 2.31349123789614063390489146856, 3.51322416617707031377994988097, 5.87244582432605089826166923091, 6.67087762072626538161918478009, 7.55224580497059979775187575187, 8.580952454804470183538270872880, 9.684759376246685710824238226498, 10.22093558811263576365367135105