| L(s) = 1 | + 3·3-s + 3·5-s − 14·7-s − 3·9-s − 15·11-s + 9·15-s + 51·17-s + 27·19-s − 42·21-s + 9·23-s − 19·25-s − 18·27-s − 12·29-s − 21·31-s − 45·33-s − 42·35-s − 31·37-s + 20·43-s − 9·45-s + 75·47-s + 147·49-s + 153·51-s + 57·53-s − 45·55-s + 81·57-s − 141·59-s − 141·61-s + ⋯ |
| L(s) = 1 | + 3-s + 3/5·5-s − 2·7-s − 1/3·9-s − 1.36·11-s + 3/5·15-s + 3·17-s + 1.42·19-s − 2·21-s + 9/23·23-s − 0.759·25-s − 2/3·27-s − 0.413·29-s − 0.677·31-s − 1.36·33-s − 6/5·35-s − 0.837·37-s + 0.465·43-s − 1/5·45-s + 1.59·47-s + 3·49-s + 3·51-s + 1.07·53-s − 0.818·55-s + 1.42·57-s − 2.38·59-s − 2.31·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.074687705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074687705\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - p T + 4 p T^{2} - p^{3} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 28 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 27 T + 604 T^{2} - 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T - 448 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 21 T + 1108 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 31 T - 408 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 75 T + 4084 T^{2} - 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 57 T + 440 T^{2} - 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 141 T + 10108 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 141 T + 10348 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 49 T - 2088 T^{2} - 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 126 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 45 T + 6004 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 73 T - 912 T^{2} - 73 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 99 T + 11188 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18050 T^{2} + p^{4} T^{4} \) |
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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
−17.06467923013066403548560008971, −16.70270169880969144905786259845, −16.04358341040107688395628229131, −15.62382139215686018694151396539, −14.80907428541092353737426324772, −14.10942743618925054272800006294, −13.64029891110490332375134093089, −13.24114509024302247940599690367, −12.29535914235251978742842726513, −12.06720739951867689665375935815, −10.44340972498096749162787822984, −10.21071167392231391345583849766, −9.369937176978151021725185616640, −9.045829347054684053544309321983, −7.62389691254039180587022414762, −7.51737015676323922492824750348, −5.76296852175578418907134492609, −5.67110042497211793025923095376, −3.28637568547457398814297236592, −3.01203156630611949535472099401,
3.01203156630611949535472099401, 3.28637568547457398814297236592, 5.67110042497211793025923095376, 5.76296852175578418907134492609, 7.51737015676323922492824750348, 7.62389691254039180587022414762, 9.045829347054684053544309321983, 9.369937176978151021725185616640, 10.21071167392231391345583849766, 10.44340972498096749162787822984, 12.06720739951867689665375935815, 12.29535914235251978742842726513, 13.24114509024302247940599690367, 13.64029891110490332375134093089, 14.10942743618925054272800006294, 14.80907428541092353737426324772, 15.62382139215686018694151396539, 16.04358341040107688395628229131, 16.70270169880969144905786259845, 17.06467923013066403548560008971
