| L(s) = 1 | − 1.35e3·3-s − 8.05e4·7-s + 1.29e6·9-s − 2.56e6·13-s + 1.06e8·19-s + 1.08e8·21-s + 4.26e7·25-s − 1.02e9·27-s − 1.33e8·31-s − 4.45e9·37-s + 3.46e9·39-s + 1.79e10·43-s − 2.28e10·49-s − 1.44e11·57-s + 8.13e10·61-s − 1.03e11·63-s + 2.42e11·67-s − 1.21e11·73-s − 5.75e10·75-s + 5.04e11·79-s + 6.98e11·81-s + 2.06e11·91-s + 1.79e11·93-s + 1.30e12·97-s + 1.99e12·103-s − 5.42e12·109-s + 6.01e12·111-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 0.684·7-s + 2.42·9-s − 0.532·13-s + 2.26·19-s + 1.26·21-s + 0.174·25-s − 2.64·27-s − 0.149·31-s − 1.73·37-s + 0.985·39-s + 2.84·43-s − 1.64·49-s − 4.19·57-s + 1.57·61-s − 1.66·63-s + 2.67·67-s − 0.805·73-s − 0.323·75-s + 2.07·79-s + 2.47·81-s + 0.364·91-s + 0.277·93-s + 1.56·97-s + 1.66·103-s − 3.23·109-s + 3.21·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+6)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.9581334030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9581334030\) |
| \(L(7)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 50 p^{3} T + p^{12} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 1706066 p^{2} T^{2} + p^{24} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 5750 p T + p^{12} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 40734109202 p^{2} T^{2} + p^{24} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 1284050 T + p^{12} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 944884986604418 T^{2} + p^{24} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 53343578 T + p^{12} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 61021177942562 p^{2} T^{2} + p^{24} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 693172036445878082 T^{2} + p^{24} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 66526202 T + p^{12} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 60235850 p T + p^{12} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 22304779456187067838 T^{2} + p^{24} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8977216250 T + p^{12} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - \)\(23\!\cdots\!78\)\( T^{2} + p^{24} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + \)\(71\!\cdots\!22\)\( T^{2} + p^{24} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - \)\(14\!\cdots\!62\)\( T^{2} + p^{24} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 40679935918 T + p^{12} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 121176846650 T + p^{12} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - \)\(30\!\cdots\!82\)\( T^{2} + p^{24} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 60956187550 T + p^{12} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 252324997702 T + p^{12} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(45\!\cdots\!38\)\( T^{2} + p^{24} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - \)\(48\!\cdots\!42\)\( T^{2} + p^{24} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 653817778850 T + p^{12} T^{2} )^{2} \) |
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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
−10.93610803472131105061516308227, −9.999758828021925510445561491168, −9.839806216506290875710143627356, −9.423869093534496612888585360335, −8.793833539684688066808107345202, −7.83847898606891089406428686226, −7.48735631304417688598171922595, −6.91781387887878662841933396416, −6.64722286129648163350997308779, −5.91981385792592211494785090662, −5.58115692082795665161865479149, −4.93844839653356353578819666989, −4.87031306488340426969794966160, −3.71582796017667069122052609329, −3.61549877800997825466109916769, −2.66041765249523946041627688295, −1.99646267182821997226021820883, −1.13606159254999472036538819149, −0.869024889047699473510334656674, −0.28559834471850539584921968889,
0.28559834471850539584921968889, 0.869024889047699473510334656674, 1.13606159254999472036538819149, 1.99646267182821997226021820883, 2.66041765249523946041627688295, 3.61549877800997825466109916769, 3.71582796017667069122052609329, 4.87031306488340426969794966160, 4.93844839653356353578819666989, 5.58115692082795665161865479149, 5.91981385792592211494785090662, 6.64722286129648163350997308779, 6.91781387887878662841933396416, 7.48735631304417688598171922595, 7.83847898606891089406428686226, 8.793833539684688066808107345202, 9.423869093534496612888585360335, 9.839806216506290875710143627356, 9.999758828021925510445561491168, 10.93610803472131105061516308227
