| L(s) = 1 | + 512·2-s + 2.89e4·3-s + 2.62e5·4-s − 7.58e6·5-s + 1.48e7·6-s + 1.05e8·7-s + 1.34e8·8-s − 3.22e8·9-s − 3.88e9·10-s − 8.39e9·11-s + 7.59e9·12-s − 3.14e10·13-s + 5.38e10·14-s − 2.19e11·15-s + 6.87e10·16-s + 4.37e9·17-s − 1.65e11·18-s + 2.25e11·19-s − 1.98e12·20-s + 3.04e12·21-s − 4.29e12·22-s + 1.24e13·23-s + 3.88e12·24-s + 3.84e13·25-s − 1.61e13·26-s − 4.30e13·27-s + 2.75e13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.849·3-s + 0.5·4-s − 1.73·5-s + 0.600·6-s + 0.985·7-s + 0.353·8-s − 0.277·9-s − 1.22·10-s − 1.07·11-s + 0.424·12-s − 0.823·13-s + 0.696·14-s − 1.47·15-s + 0.250·16-s + 0.00894·17-s − 0.196·18-s + 0.160·19-s − 0.868·20-s + 0.837·21-s − 0.758·22-s + 1.44·23-s + 0.300·24-s + 2.01·25-s − 0.582·26-s − 1.08·27-s + 0.492·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(3.183098087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.183098087\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 512T \) |
| 29 | \( 1 - 1.45e13T \) |
| good | 3 | \( 1 - 2.89e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 7.58e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.05e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 8.39e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 3.14e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 4.37e9T + 2.39e23T^{2} \) |
| 19 | \( 1 - 2.25e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 1.24e13T + 7.46e25T^{2} \) |
| 31 | \( 1 - 1.89e14T + 2.16e28T^{2} \) |
| 37 | \( 1 + 1.23e15T + 6.24e29T^{2} \) |
| 41 | \( 1 - 3.86e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 1.26e15T + 1.08e31T^{2} \) |
| 47 | \( 1 - 3.14e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 3.92e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 3.05e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 3.51e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.82e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 4.52e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 2.56e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 1.72e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 1.47e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 5.37e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 2.38e18T + 5.60e37T^{2} \) |
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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
−11.55982162443489225309047038652, −10.66412942774375993131008195703, −8.736958171685864060414855044200, −7.86656461854809115832000195874, −7.27446644550018815677341914163, −5.21077297685699605601441388808, −4.38070489443006508639212811459, −3.21291361850930841183871797948, −2.42366194278302650949205536554, −0.68185115856847051460238549509,
0.68185115856847051460238549509, 2.42366194278302650949205536554, 3.21291361850930841183871797948, 4.38070489443006508639212811459, 5.21077297685699605601441388808, 7.27446644550018815677341914163, 7.86656461854809115832000195874, 8.736958171685864060414855044200, 10.66412942774375993131008195703, 11.55982162443489225309047038652
