| L(s) = 1 | − 52.2·2-s − 945.·3-s − 5.46e3·4-s − 3.07e4·5-s + 4.93e4·6-s − 5.24e5·7-s + 7.13e5·8-s − 7.00e5·9-s + 1.60e6·10-s − 5.64e6·11-s + 5.16e6·12-s + 4.82e6·13-s + 2.73e7·14-s + 2.90e7·15-s + 7.55e6·16-s − 6.37e6·17-s + 3.65e7·18-s + 2.63e8·19-s + 1.67e8·20-s + 4.95e8·21-s + 2.94e8·22-s − 5.77e8·23-s − 6.74e8·24-s − 2.76e8·25-s − 2.51e8·26-s + 2.16e9·27-s + 2.86e9·28-s + ⋯ |
| L(s) = 1 | − 0.576·2-s − 0.748·3-s − 0.667·4-s − 0.879·5-s + 0.431·6-s − 1.68·7-s + 0.961·8-s − 0.439·9-s + 0.507·10-s − 0.960·11-s + 0.499·12-s + 0.277·13-s + 0.971·14-s + 0.658·15-s + 0.112·16-s − 0.0640·17-s + 0.253·18-s + 1.28·19-s + 0.586·20-s + 1.26·21-s + 0.554·22-s − 0.814·23-s − 0.720·24-s − 0.226·25-s − 0.159·26-s + 1.07·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.1088666180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1088666180\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 4.82e6T \) |
| good | 2 | \( 1 + 52.2T + 8.19e3T^{2} \) |
| 3 | \( 1 + 945.T + 1.59e6T^{2} \) |
| 5 | \( 1 + 3.07e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.24e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.64e6T + 3.45e13T^{2} \) |
| 17 | \( 1 + 6.37e6T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.63e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 5.77e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 5.22e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.99e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 3.00e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.79e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 4.83e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 6.36e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.74e10T + 2.60e22T^{2} \) |
| 59 | \( 1 + 1.44e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 3.68e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.25e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 2.91e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.58e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 6.22e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.30e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.28e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 3.69e12T + 6.73e25T^{2} \) |
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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
−16.58891043952810007534868712664, −15.80194584380202190455480249081, −13.57635400522933660581432316308, −12.26985276412525168079224798474, −10.63704516153393940497543439681, −9.255074273361520064740555282017, −7.55526288924803430480599310697, −5.59876597396294778306212505117, −3.55528473979559362000336088663, −0.26494562157779490518252433396,
0.26494562157779490518252433396, 3.55528473979559362000336088663, 5.59876597396294778306212505117, 7.55526288924803430480599310697, 9.255074273361520064740555282017, 10.63704516153393940497543439681, 12.26985276412525168079224798474, 13.57635400522933660581432316308, 15.80194584380202190455480249081, 16.58891043952810007534868712664
