| L(s) = 1 | + 16·2-s − 156·3-s + 256·4-s + 870·5-s − 2.49e3·6-s − 952·7-s + 4.09e3·8-s + 4.65e3·9-s + 1.39e4·10-s − 5.61e4·11-s − 3.99e4·12-s + 1.78e5·13-s − 1.52e4·14-s − 1.35e5·15-s + 6.55e4·16-s − 2.47e5·17-s + 7.44e4·18-s + 3.15e5·19-s + 2.22e5·20-s + 1.48e5·21-s − 8.98e5·22-s + 2.04e5·23-s − 6.38e5·24-s − 1.19e6·25-s + 2.84e6·26-s + 2.34e6·27-s − 2.43e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.11·3-s + 1/2·4-s + 0.622·5-s − 0.786·6-s − 0.149·7-s + 0.353·8-s + 0.236·9-s + 0.440·10-s − 1.15·11-s − 0.555·12-s + 1.72·13-s − 0.105·14-s − 0.692·15-s + 1/4·16-s − 0.719·17-s + 0.167·18-s + 0.555·19-s + 0.311·20-s + 0.166·21-s − 0.817·22-s + 0.152·23-s − 0.393·24-s − 0.612·25-s + 1.22·26-s + 0.849·27-s − 0.0749·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.141162785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.141162785\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 + 52 p T + p^{9} T^{2} \) |
| 5 | \( 1 - 174 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 136 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 56148 T + p^{9} T^{2} \) |
| 13 | \( 1 - 178094 T + p^{9} T^{2} \) |
| 17 | \( 1 + 247662 T + p^{9} T^{2} \) |
| 19 | \( 1 - 315380 T + p^{9} T^{2} \) |
| 23 | \( 1 - 204504 T + p^{9} T^{2} \) |
| 29 | \( 1 + 3840450 T + p^{9} T^{2} \) |
| 31 | \( 1 + 1309408 T + p^{9} T^{2} \) |
| 37 | \( 1 - 4307078 T + p^{9} T^{2} \) |
| 41 | \( 1 - 1512042 T + p^{9} T^{2} \) |
| 43 | \( 1 - 33670604 T + p^{9} T^{2} \) |
| 47 | \( 1 + 10581072 T + p^{9} T^{2} \) |
| 53 | \( 1 - 16616214 T + p^{9} T^{2} \) |
| 59 | \( 1 - 112235100 T + p^{9} T^{2} \) |
| 61 | \( 1 + 33197218 T + p^{9} T^{2} \) |
| 67 | \( 1 + 121372252 T + p^{9} T^{2} \) |
| 71 | \( 1 + 387172728 T + p^{9} T^{2} \) |
| 73 | \( 1 - 255240074 T + p^{9} T^{2} \) |
| 79 | \( 1 - 492101840 T + p^{9} T^{2} \) |
| 83 | \( 1 + 457420236 T + p^{9} T^{2} \) |
| 89 | \( 1 + 31809510 T + p^{9} T^{2} \) |
| 97 | \( 1 + 673532062 T + p^{9} T^{2} \) |
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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
−28.35318874818419914747004790260, −25.79352224805499530229115622654, −23.76815125200101476947248485954, −22.54028217487831735387166041105, −20.92743840696366585639188931061, −17.98823119051954732852794147681, −16.04530025584379928185207493633, −13.28647941010376044614841610393, −11.03470393984095732596654437189, −5.81514644481805891097876856886,
5.81514644481805891097876856886, 11.03470393984095732596654437189, 13.28647941010376044614841610393, 16.04530025584379928185207493633, 17.98823119051954732852794147681, 20.92743840696366585639188931061, 22.54028217487831735387166041105, 23.76815125200101476947248485954, 25.79352224805499530229115622654, 28.35318874818419914747004790260
