| L(s) = 1 | + 9.10·2-s − 81·3-s − 429.·4-s + 625·5-s − 737.·6-s + 6.71e3·7-s − 8.56e3·8-s + 6.56e3·9-s + 5.68e3·10-s + 7.38e4·11-s + 3.47e4·12-s + 4.82e4·13-s + 6.11e4·14-s − 5.06e4·15-s + 1.41e5·16-s + 3.02e5·17-s + 5.97e4·18-s − 1.30e5·19-s − 2.68e5·20-s − 5.43e5·21-s + 6.72e5·22-s + 2.35e6·23-s + 6.93e5·24-s + 3.90e5·25-s + 4.38e5·26-s − 5.31e5·27-s − 2.88e6·28-s + ⋯ |
| L(s) = 1 | + 0.402·2-s − 0.577·3-s − 0.838·4-s + 0.447·5-s − 0.232·6-s + 1.05·7-s − 0.739·8-s + 0.333·9-s + 0.179·10-s + 1.52·11-s + 0.483·12-s + 0.468·13-s + 0.425·14-s − 0.258·15-s + 0.540·16-s + 0.877·17-s + 0.134·18-s − 0.229·19-s − 0.374·20-s − 0.610·21-s + 0.611·22-s + 1.75·23-s + 0.426·24-s + 0.200·25-s + 0.188·26-s − 0.192·27-s − 0.885·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.890883092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.890883092\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 - 9.10T + 512T^{2} \) |
| 7 | \( 1 - 6.71e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.38e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.82e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.02e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 2.35e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.23e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.87e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.75e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 9.93e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.79e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.32e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.12e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.19e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.12e4T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.64e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.51e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.11e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.07e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.22e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.96e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.27e9T + 7.60e17T^{2} \) |
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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
−10.29298907841265842845413460014, −9.196698895472659940372242563672, −8.585829198020997893291791125462, −7.21315310107207929948630882173, −6.06633058662662722983880622504, −5.23139116087686483720141526393, −4.39980707991666001276822078537, −3.38075638818063455821643929124, −1.55651129063327280765132020012, −0.833014284840587501281655905950,
0.833014284840587501281655905950, 1.55651129063327280765132020012, 3.38075638818063455821643929124, 4.39980707991666001276822078537, 5.23139116087686483720141526393, 6.06633058662662722983880622504, 7.21315310107207929948630882173, 8.585829198020997893291791125462, 9.196698895472659940372242563672, 10.29298907841265842845413460014
