| 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
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.2275425459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2275425459\) |
| \(L(7)\) |
|
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
−10.52455267081732254492184897282, −9.988303947409853992090359791462, −9.655220587390846557133239924117, −8.795659100685237043244394801830, −8.700022518656930416334865683243, −8.203078579693832651079659813077, −7.935658520307721413448240208570, −7.02359601612854975687842466386, −6.94749324143603647906461061395, −6.25617383505587416209330192238, −5.36049959596152217573496932714, −4.55819252915014169925013905827, −4.55723672393260592531309261231, −3.70446632584424943642504720976, −3.27043672787105469613004731149, −2.65900580571156517887042668767, −2.13892189079284104357672774902, −1.56726631729981531498248874654, −1.44064962709756428002746306175, −0.06714329032128252254852585785,
0.06714329032128252254852585785, 1.44064962709756428002746306175, 1.56726631729981531498248874654, 2.13892189079284104357672774902, 2.65900580571156517887042668767, 3.27043672787105469613004731149, 3.70446632584424943642504720976, 4.55723672393260592531309261231, 4.55819252915014169925013905827, 5.36049959596152217573496932714, 6.25617383505587416209330192238, 6.94749324143603647906461061395, 7.02359601612854975687842466386, 7.935658520307721413448240208570, 8.203078579693832651079659813077, 8.700022518656930416334865683243, 8.795659100685237043244394801830, 9.655220587390846557133239924117, 9.988303947409853992090359791462, 10.52455267081732254492184897282
