| L(s) = 1 | + 1.96e4·3-s − 1.31e5·4-s − 2.75e6·7-s + 2.58e8·9-s − 2.57e9·12-s − 2.53e11·19-s − 5.42e10·21-s + 7.62e11·25-s + 2.54e12·27-s + 3.61e11·28-s − 8.74e12·31-s − 3.38e13·36-s − 4.21e13·37-s − 1.09e14·43-s − 2.25e14·49-s − 4.99e15·57-s − 5.18e15·61-s − 7.12e14·63-s + 2.25e15·64-s + 2.69e14·67-s + 1.49e16·73-s + 1.50e16·75-s + 3.32e16·76-s + 2.64e16·79-s + 1.66e16·81-s + 7.11e15·84-s − 1.72e17·93-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s − 0.180·7-s + 2·9-s − 1.73·12-s − 3.43·19-s − 0.313·21-s + 25-s + 1.73·27-s + 0.180·28-s − 1.84·31-s − 2·36-s − 1.97·37-s − 1.43·43-s − 0.967·49-s − 5.94·57-s − 3.46·61-s − 0.361·63-s + 64-s + 0.0810·67-s + 2.17·73-s + 1.73·75-s + 3.43·76-s + 1.96·79-s + 81-s + 0.313·84-s − 3.19·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.7397357773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7397357773\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p^{9} T + p^{17} T^{2} \) |
| 7 | $C_2$ | \( 1 + 2758181 T + p^{17} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p^{17} T^{2} + p^{34} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p^{17} T^{2} + p^{34} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{17} T^{2} + p^{34} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3266839877 T + p^{17} T^{2} )( 1 + 3266839877 T + p^{17} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p^{17} T^{2} + p^{34} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 116632805768 T + p^{17} T^{2} )( 1 + 137302173991 T + p^{17} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p^{17} T^{2} + p^{34} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{17} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 351446059196 T + p^{17} T^{2} )( 1 + 8395073038759 T + p^{17} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 14723537266559 T + p^{17} T^{2} )( 1 + 27378365365430 T + p^{17} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p^{17} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 54897704698835 T + p^{17} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p^{17} T^{2} + p^{34} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{17} T^{2} + p^{34} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p^{17} T^{2} + p^{34} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2528437643672447 T + p^{17} T^{2} )( 1 + 2653755538627021 T + p^{17} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5886945232004336 T + p^{17} T^{2} )( 1 + 5617606043215765 T + p^{17} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p^{17} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12856663699194230 T + p^{17} T^{2} )( 1 - 2131541818135687 T + p^{17} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17716682078842556 T + p^{17} T^{2} )( 1 - 8750552126467667 T + p^{17} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{17} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p^{17} T^{2} + p^{34} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 118297695891443726 T + p^{17} T^{2} )( 1 + 118297695891443726 T + p^{17} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.52442582796849976908594628050, −13.74031202303732162307301646737, −13.43189344608350551498973227791, −12.58127377865421462491688527886, −12.52906423313397648282384532445, −10.72987156030919159941267717933, −10.62002563956613514908130527719, −9.334461195804108735254837475357, −9.212154440367386356237567623329, −8.311901461894302737167807047891, −8.276911895899791172137384170713, −6.99244850594775641708476934736, −6.47156239331448620788643065646, −5.06743510926847890243346793662, −4.37742455486164328761981449108, −3.77792064457441427014083380539, −3.09718009268942359849182785808, −2.04859868907675581650995899585, −1.71783164558578473133297397705, −0.21030744653775516846655122657,
0.21030744653775516846655122657, 1.71783164558578473133297397705, 2.04859868907675581650995899585, 3.09718009268942359849182785808, 3.77792064457441427014083380539, 4.37742455486164328761981449108, 5.06743510926847890243346793662, 6.47156239331448620788643065646, 6.99244850594775641708476934736, 8.276911895899791172137384170713, 8.311901461894302737167807047891, 9.212154440367386356237567623329, 9.334461195804108735254837475357, 10.62002563956613514908130527719, 10.72987156030919159941267717933, 12.52906423313397648282384532445, 12.58127377865421462491688527886, 13.43189344608350551498973227791, 13.74031202303732162307301646737, 14.52442582796849976908594628050
