| L(s) = 1 | − 21.9·2-s − 1.18e4·3-s − 1.30e5·4-s + 1.39e6·5-s + 2.60e5·6-s − 1.09e7·7-s + 5.74e6·8-s + 1.19e7·9-s − 3.06e7·10-s + 1.26e9·11-s + 1.55e9·12-s + 7.40e8·13-s + 2.40e8·14-s − 1.65e10·15-s + 1.69e10·16-s + 2.85e10·17-s − 2.63e8·18-s − 3.84e10·19-s − 1.82e11·20-s + 1.29e11·21-s − 2.78e10·22-s − 4.52e11·23-s − 6.82e10·24-s + 1.18e12·25-s − 1.62e10·26-s + 1.39e12·27-s + 1.42e12·28-s + ⋯ |
| L(s) = 1 | − 0.0606·2-s − 1.04·3-s − 0.996·4-s + 1.59·5-s + 0.0634·6-s − 0.716·7-s + 0.121·8-s + 0.0928·9-s − 0.0969·10-s + 1.78·11-s + 1.04·12-s + 0.251·13-s + 0.0434·14-s − 1.67·15-s + 0.988·16-s + 0.992·17-s − 0.00563·18-s − 0.519·19-s − 1.59·20-s + 0.749·21-s − 0.108·22-s − 1.20·23-s − 0.126·24-s + 1.55·25-s − 0.0152·26-s + 0.948·27-s + 0.714·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.449768200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449768200\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 - 1.46e14T \) |
| good | 2 | \( 1 + 21.9T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.18e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.39e6T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.09e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.26e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 7.40e8T + 8.65e18T^{2} \) |
| 17 | \( 1 - 2.85e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 3.84e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.52e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.97e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 6.32e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.88e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 2.19e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 4.02e12T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.93e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.49e14T + 2.05e29T^{2} \) |
| 61 | \( 1 - 2.03e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.10e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 7.15e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 3.38e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.92e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.90e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 2.64e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.10e17T + 5.95e33T^{2} \) |
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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
−11.80504226358305270142191285850, −10.20373587584269819506034331856, −9.669272037866256731036547821356, −8.626342692158441801476822668923, −6.40979415986518127351377160214, −6.03059869182869062954854226485, −4.88396384706988429164795896196, −3.48575047692029836862896328990, −1.62923730291803969610080844776, −0.65288605712092681914503385598,
0.65288605712092681914503385598, 1.62923730291803969610080844776, 3.48575047692029836862896328990, 4.88396384706988429164795896196, 6.03059869182869062954854226485, 6.40979415986518127351377160214, 8.626342692158441801476822668923, 9.669272037866256731036547821356, 10.20373587584269819506034331856, 11.80504226358305270142191285850
