| L(s) = 1 | − 536.·2-s − 1.96e4·3-s − 2.36e5·4-s − 6.83e5·5-s + 1.05e7·6-s + 1.13e8·7-s + 4.08e8·8-s + 3.87e8·9-s + 3.66e8·10-s + 6.43e9·11-s + 4.65e9·12-s − 5.75e10·13-s − 6.09e10·14-s + 1.34e10·15-s − 9.47e10·16-s + 6.85e11·17-s − 2.07e11·18-s + 1.22e12·19-s + 1.61e11·20-s − 2.23e12·21-s − 3.44e12·22-s + 5.02e12·23-s − 8.03e12·24-s − 1.86e13·25-s + 3.08e13·26-s − 7.62e12·27-s − 2.69e13·28-s + ⋯ |
| L(s) = 1 | − 0.740·2-s − 0.577·3-s − 0.451·4-s − 0.156·5-s + 0.427·6-s + 1.06·7-s + 1.07·8-s + 0.333·9-s + 0.115·10-s + 0.822·11-s + 0.260·12-s − 1.50·13-s − 0.788·14-s + 0.0903·15-s − 0.344·16-s + 1.40·17-s − 0.246·18-s + 0.869·19-s + 0.0706·20-s − 0.614·21-s − 0.609·22-s + 0.581·23-s − 0.620·24-s − 0.975·25-s + 1.11·26-s − 0.192·27-s − 0.480·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.8734009930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8734009930\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.96e4T \) |
| good | 2 | \( 1 + 536.T + 5.24e5T^{2} \) |
| 5 | \( 1 + 6.83e5T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.13e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 6.43e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 5.75e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 6.85e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.22e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 5.02e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.53e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 3.92e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.35e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 5.75e14T + 4.39e30T^{2} \) |
| 43 | \( 1 - 3.36e14T + 1.08e31T^{2} \) |
| 47 | \( 1 + 6.99e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 1.69e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 2.70e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 5.11e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 2.07e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 2.35e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 7.43e16T + 2.53e35T^{2} \) |
| 79 | \( 1 + 6.98e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 3.25e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 1.43e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.66e18T + 5.60e37T^{2} \) |
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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
−21.70026776405241247383190315253, −19.50800368651085823387801775358, −17.90021638358707200352573178688, −16.85496353188924852403156201379, −14.36215004365440983920762826177, −11.86029359705413426139138588874, −9.845666150611256034314878520095, −7.75778586459758829259083930184, −4.82452618151653980702153629445, −1.03035876013938774134787752717,
1.03035876013938774134787752717, 4.82452618151653980702153629445, 7.75778586459758829259083930184, 9.845666150611256034314878520095, 11.86029359705413426139138588874, 14.36215004365440983920762826177, 16.85496353188924852403156201379, 17.90021638358707200352573178688, 19.50800368651085823387801775358, 21.70026776405241247383190315253
