| L(s) = 1 | − 1.02e3·2-s − 6.26e4·3-s + 1.04e6·4-s + 6.41e7·6-s + 1.35e9·7-s − 1.07e9·8-s − 6.53e9·9-s + 6.02e10·11-s − 6.56e10·12-s + 6.48e11·13-s − 1.39e12·14-s + 1.09e12·16-s − 1.36e12·17-s + 6.69e12·18-s + 3.73e13·19-s − 8.50e13·21-s − 6.16e13·22-s + 3.31e14·23-s + 6.72e13·24-s − 6.64e14·26-s + 1.06e15·27-s + 1.42e15·28-s + 2.94e15·29-s + 6.11e15·31-s − 1.12e15·32-s − 3.77e15·33-s + 1.40e15·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.612·3-s + 0.5·4-s + 0.432·6-s + 1.81·7-s − 0.353·8-s − 0.625·9-s + 0.699·11-s − 0.306·12-s + 1.30·13-s − 1.28·14-s + 0.250·16-s − 0.164·17-s + 0.442·18-s + 1.39·19-s − 1.11·21-s − 0.494·22-s + 1.66·23-s + 0.216·24-s − 0.922·26-s + 0.994·27-s + 0.908·28-s + 1.30·29-s + 1.33·31-s − 0.176·32-s − 0.428·33-s + 0.116·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.179184044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179184044\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.02e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 6.26e4T + 1.04e10T^{2} \) |
| 7 | \( 1 - 1.35e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 6.02e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 6.48e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.36e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.73e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.31e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.94e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.11e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.13e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.53e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.11e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.03e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.33e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 8.63e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.07e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.05e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.84e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 8.35e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.13e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.83e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.15e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.08e20T + 5.27e41T^{2} \) |
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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
−11.44381877787588475228662813703, −10.52665704387963265454512953656, −8.872150822527220218434834249826, −8.236884954869106777714465494292, −6.90022787906107811904833961560, −5.66155733786185090493260809070, −4.64250606391870930317138077826, −2.97164834377977849841012288512, −1.34821863283083695095565571867, −0.930385705593866210752909475586,
0.930385705593866210752909475586, 1.34821863283083695095565571867, 2.97164834377977849841012288512, 4.64250606391870930317138077826, 5.66155733786185090493260809070, 6.90022787906107811904833961560, 8.236884954869106777714465494292, 8.872150822527220218434834249826, 10.52665704387963265454512953656, 11.44381877787588475228662813703
