| L(s) = 1 | − 1.02e3·2-s + 5.90e4·3-s + 1.04e6·4-s + 2.64e7·5-s − 6.04e7·6-s + 1.66e8·7-s − 1.07e9·8-s + 3.48e9·9-s − 2.70e10·10-s − 1.04e11·11-s + 6.19e10·12-s + 3.35e11·13-s − 1.70e11·14-s + 1.56e12·15-s + 1.09e12·16-s + 1.45e13·17-s − 3.57e12·18-s + 3.56e12·19-s + 2.77e13·20-s + 9.80e12·21-s + 1.07e14·22-s + 2.22e14·23-s − 6.34e13·24-s + 2.22e14·25-s − 3.43e14·26-s + 2.05e14·27-s + 1.74e14·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.21·5-s − 0.408·6-s + 0.222·7-s − 0.353·8-s + 1/3·9-s − 0.856·10-s − 1.21·11-s + 0.288·12-s + 0.675·13-s − 0.157·14-s + 0.699·15-s + 1/4·16-s + 1.75·17-s − 0.235·18-s + 0.133·19-s + 0.605·20-s + 0.128·21-s + 0.862·22-s + 1.11·23-s − 0.204·24-s + 0.466·25-s − 0.477·26-s + 0.192·27-s + 0.111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.113543625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113543625\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{10} T \) |
| 3 | \( 1 - p^{10} T \) |
| good | 5 | \( 1 - 1057782 p^{2} T + p^{21} T^{2} \) |
| 7 | \( 1 - 166115864 T + p^{21} T^{2} \) |
| 11 | \( 1 + 104878761780 T + p^{21} T^{2} \) |
| 13 | \( 1 - 25814717366 p T + p^{21} T^{2} \) |
| 17 | \( 1 - 858596750802 p T + p^{21} T^{2} \) |
| 19 | \( 1 - 187869998684 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 222369240588600 T + p^{21} T^{2} \) |
| 29 | \( 1 - 2194109701319454 T + p^{21} T^{2} \) |
| 31 | \( 1 + 8723627187590032 T + p^{21} T^{2} \) |
| 37 | \( 1 - 37470891663324758 T + p^{21} T^{2} \) |
| 41 | \( 1 - 86616741616565034 T + p^{21} T^{2} \) |
| 43 | \( 1 - 131416928813078444 T + p^{21} T^{2} \) |
| 47 | \( 1 - 339041180377015440 T + p^{21} T^{2} \) |
| 53 | \( 1 + 1571494796445297834 T + p^{21} T^{2} \) |
| 59 | \( 1 - 5232984701774509020 T + p^{21} T^{2} \) |
| 61 | \( 1 + 4788384962739867250 T + p^{21} T^{2} \) |
| 67 | \( 1 + 15480328743911983516 T + p^{21} T^{2} \) |
| 71 | \( 1 + 12930906477499746840 T + p^{21} T^{2} \) |
| 73 | \( 1 + 44257184658687636502 T + p^{21} T^{2} \) |
| 79 | \( 1 + 14888578935758942752 T + p^{21} T^{2} \) |
| 83 | \( 1 - 37085068910999181588 T + p^{21} T^{2} \) |
| 89 | \( 1 + \)\(10\!\cdots\!42\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 - \)\(13\!\cdots\!98\)\( T + p^{21} T^{2} \) |
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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
−17.82421804080436720314911140248, −16.25507430953519897627229315723, −14.47720876073577057472175971798, −12.98881245749964730187319475597, −10.59939879585078710501901450384, −9.335115698246033359333770225560, −7.72896906974448705492811639355, −5.65277233132863441367195096616, −2.79409909226197795371860950223, −1.27773392921501138795686175163,
1.27773392921501138795686175163, 2.79409909226197795371860950223, 5.65277233132863441367195096616, 7.72896906974448705492811639355, 9.335115698246033359333770225560, 10.59939879585078710501901450384, 12.98881245749964730187319475597, 14.47720876073577057472175971798, 16.25507430953519897627229315723, 17.82421804080436720314911140248
