| L(s) = 1 | − 1.02e3·2-s + 1.63e5·3-s + 1.04e6·4-s − 1.67e8·6-s − 1.27e8·7-s − 1.07e9·8-s + 1.64e10·9-s + 1.26e10·11-s + 1.71e11·12-s + 3.89e11·13-s + 1.30e11·14-s + 1.09e12·16-s + 6.14e12·17-s − 1.67e13·18-s + 4.71e13·19-s − 2.08e13·21-s − 1.29e13·22-s − 2.22e14·23-s − 1.75e14·24-s − 3.99e14·26-s + 9.73e14·27-s − 1.33e14·28-s − 1.52e15·29-s − 5.97e15·31-s − 1.12e15·32-s + 2.07e15·33-s − 6.29e15·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.60·3-s + 0.5·4-s − 1.13·6-s − 0.170·7-s − 0.353·8-s + 1.56·9-s + 0.147·11-s + 0.801·12-s + 0.784·13-s + 0.120·14-s + 0.250·16-s + 0.739·17-s − 1.10·18-s + 1.76·19-s − 0.272·21-s − 0.103·22-s − 1.12·23-s − 0.566·24-s − 0.554·26-s + 0.910·27-s − 0.0851·28-s − 0.674·29-s − 1.30·31-s − 0.176·32-s + 0.235·33-s − 0.523·34-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
| \(L(11)\) |
\(\approx\) |
\(3.482752135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.482752135\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 1.63e5T + 1.04e10T^{2} \) |
| 7 | \( 1 + 1.27e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.26e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 3.89e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 6.14e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.71e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.22e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.52e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 5.97e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.33e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.52e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.62e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.41e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 4.40e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.30e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 5.54e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 4.60e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.90e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.14e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.49e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.68e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.29e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.45e20T + 5.27e41T^{2} \) |
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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
−11.18429639357060965886998269695, −9.720687068206702609759868505733, −9.230931974378937291445842616592, −8.015176359243307971489822118774, −7.43330836836447757081034532858, −5.83291957898879035963299413351, −3.86531803198152086218551593765, −3.05575751006884189435176829278, −1.92275602464266773817280822177, −0.901088590241127066286705778149,
0.901088590241127066286705778149, 1.92275602464266773817280822177, 3.05575751006884189435176829278, 3.86531803198152086218551593765, 5.83291957898879035963299413351, 7.43330836836447757081034532858, 8.015176359243307971489822118774, 9.230931974378937291445842616592, 9.720687068206702609759868505733, 11.18429639357060965886998269695
