| L(s) = 1 | + 43.3·2-s + 81·3-s + 1.36e3·4-s + 2.36e3·5-s + 3.51e3·6-s − 2.16e3·7-s + 3.71e4·8-s + 6.56e3·9-s + 1.02e5·10-s − 4.65e4·11-s + 1.10e5·12-s − 2.93e4·13-s − 9.37e4·14-s + 1.91e5·15-s + 9.10e5·16-s − 3.36e5·17-s + 2.84e5·18-s + 6.88e5·19-s + 3.23e6·20-s − 1.75e5·21-s − 2.01e6·22-s + 1.45e6·23-s + 3.01e6·24-s + 3.62e6·25-s − 1.27e6·26-s + 5.31e5·27-s − 2.95e6·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.577·3-s + 2.67·4-s + 1.69·5-s + 1.10·6-s − 0.340·7-s + 3.20·8-s + 0.333·9-s + 3.23·10-s − 0.958·11-s + 1.54·12-s − 0.285·13-s − 0.652·14-s + 0.975·15-s + 3.47·16-s − 0.976·17-s + 0.638·18-s + 1.21·19-s + 4.51·20-s − 0.196·21-s − 1.83·22-s + 1.08·23-s + 1.85·24-s + 1.85·25-s − 0.546·26-s + 0.192·27-s − 0.909·28-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
| \(L(5)\) |
\(\approx\) |
\(12.85901940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.85901940\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 43.3T + 512T^{2} \) |
| 5 | \( 1 - 2.36e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.16e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.93e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.36e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.45e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.52e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.65e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.29e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.92e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.67e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.71e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.13e8T + 3.29e15T^{2} \) |
| 61 | \( 1 + 2.96e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.00e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 8.90e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.68e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.37e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.59e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.74e8T + 7.60e17T^{2} \) |
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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
−11.13586736648626718953120198215, −10.27150297380405423558013447731, −9.196540850994065904147432694578, −7.41101070861927250623736837025, −6.50588481112312010775444003420, −5.48959488026216035458439407362, −4.82489507174976390440764099350, −3.22620868517848746252749113709, −2.51555904440685158269258858874, −1.60092418541279185467575921379,
1.60092418541279185467575921379, 2.51555904440685158269258858874, 3.22620868517848746252749113709, 4.82489507174976390440764099350, 5.48959488026216035458439407362, 6.50588481112312010775444003420, 7.41101070861927250623736837025, 9.196540850994065904147432694578, 10.27150297380405423558013447731, 11.13586736648626718953120198215
