| L(s) = 1 | − 512·2-s + 1.96e4·3-s + 2.62e5·4-s − 5.84e6·5-s − 1.00e7·6-s + 1.73e8·7-s − 1.34e8·8-s + 3.87e8·9-s + 2.99e9·10-s − 7.31e9·11-s + 5.15e9·12-s − 4.18e10·13-s − 8.88e10·14-s − 1.15e11·15-s + 6.87e10·16-s − 9.58e10·17-s − 1.98e11·18-s − 2.41e12·19-s − 1.53e12·20-s + 3.41e12·21-s + 3.74e12·22-s − 1.32e13·23-s − 2.64e12·24-s + 1.51e13·25-s + 2.14e13·26-s + 7.62e12·27-s + 4.54e13·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.33·5-s − 0.408·6-s + 1.62·7-s − 0.353·8-s + 1/3·9-s + 0.947·10-s − 0.935·11-s + 0.288·12-s − 1.09·13-s − 1.14·14-s − 0.773·15-s + 1/4·16-s − 0.196·17-s − 0.235·18-s − 1.71·19-s − 0.669·20-s + 0.938·21-s + 0.661·22-s − 1.53·23-s − 0.204·24-s + 0.793·25-s + 0.773·26-s + 0.192·27-s + 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{9} T \) |
| 3 | \( 1 - p^{9} T \) |
| good | 5 | \( 1 + 1169898 p T + p^{19} T^{2} \) |
| 7 | \( 1 - 3541448 p^{2} T + p^{19} T^{2} \) |
| 11 | \( 1 + 664802580 p T + p^{19} T^{2} \) |
| 13 | \( 1 + 41845065034 T + p^{19} T^{2} \) |
| 17 | \( 1 + 95834399598 T + p^{19} T^{2} \) |
| 19 | \( 1 + 2419072521316 T + p^{19} T^{2} \) |
| 23 | \( 1 + 13218544831800 T + p^{19} T^{2} \) |
| 29 | \( 1 - 22096708325526 T + p^{19} T^{2} \) |
| 31 | \( 1 - 54205000762928 T + p^{19} T^{2} \) |
| 37 | \( 1 + 754675410892066 T + p^{19} T^{2} \) |
| 41 | \( 1 - 1015505924861274 T + p^{19} T^{2} \) |
| 43 | \( 1 - 2307401507879108 T + p^{19} T^{2} \) |
| 47 | \( 1 - 73656034083120 T + p^{19} T^{2} \) |
| 53 | \( 1 + 5772296141217378 T + p^{19} T^{2} \) |
| 59 | \( 1 + 129569039139755820 T + p^{19} T^{2} \) |
| 61 | \( 1 - 114049208167000550 T + p^{19} T^{2} \) |
| 67 | \( 1 + 131076909617853748 T + p^{19} T^{2} \) |
| 71 | \( 1 - 532256691369812760 T + p^{19} T^{2} \) |
| 73 | \( 1 + 801680088264316774 T + p^{19} T^{2} \) |
| 79 | \( 1 + 799386550683767488 T + p^{19} T^{2} \) |
| 83 | \( 1 - 1021049179204582236 T + p^{19} T^{2} \) |
| 89 | \( 1 - 4064785168167821322 T + p^{19} T^{2} \) |
| 97 | \( 1 - 11014962791774968034 T + p^{19} T^{2} \) |
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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
−17.53747408365542802859143576299, −15.66363467190436668403410752822, −14.60187008479543226580526396555, −12.10450310459711818102730558881, −10.66525859094654714659506264435, −8.347367857816285804366512846663, −7.64307317874056507155696621606, −4.44834237788697921626535666084, −2.16354679626591636725421605164, 0,
2.16354679626591636725421605164, 4.44834237788697921626535666084, 7.64307317874056507155696621606, 8.347367857816285804366512846663, 10.66525859094654714659506264435, 12.10450310459711818102730558881, 14.60187008479543226580526396555, 15.66363467190436668403410752822, 17.53747408365542802859143576299
