| L(s) = 1 | + 4.09e3·4-s + 1.25e7·16-s − 4.11e7·19-s + 5.92e8·31-s − 1.95e9·49-s − 2.59e10·61-s + 3.43e10·64-s − 1.68e11·76-s − 6.57e10·79-s − 5.15e9·109-s − 5.70e11·121-s + 2.42e12·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.47e12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s − 3.81·19-s + 3.71·31-s − 0.989·49-s − 3.93·61-s + 4·64-s − 7.62·76-s − 2.40·79-s − 0.0321·109-s − 2·121-s + 7.43·124-s − 0.821·169-s − 1.97·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.936084947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936084947\) |
| \(L(\frac{13}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 1956649739 T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1471948630151 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 20581901 T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 296476943 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 257128060064352074 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 1268503014678232811 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12977292913 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - \)\(20\!\cdots\!41\)\( T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - \)\(23\!\cdots\!54\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 32885832404 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - \)\(11\!\cdots\!81\)\( T^{2} + p^{22} T^{4} \) |
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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
−10.72398660732094554317284377299, −10.22430495257744996009965708671, −9.866982948581862354656193353790, −8.864460747184879980214115418809, −8.476212218539954738376133982188, −8.009323697681412739780083860085, −7.63612197043350794404737402345, −6.86605712087388735893871501233, −6.41516331335451672701692991377, −6.25166101264083373855331701155, −5.96030100461467644500069452885, −4.80685039269435234923315953905, −4.47278923207750222464459056639, −3.82069937032546990998985706582, −2.98188726228109568830361848490, −2.55122520629527507726328797311, −2.32466118015015095259094436643, −1.42856583045464094732493809850, −1.33825438559568423726978602347, −0.22569382429528437715592280617,
0.22569382429528437715592280617, 1.33825438559568423726978602347, 1.42856583045464094732493809850, 2.32466118015015095259094436643, 2.55122520629527507726328797311, 2.98188726228109568830361848490, 3.82069937032546990998985706582, 4.47278923207750222464459056639, 4.80685039269435234923315953905, 5.96030100461467644500069452885, 6.25166101264083373855331701155, 6.41516331335451672701692991377, 6.86605712087388735893871501233, 7.63612197043350794404737402345, 8.009323697681412739780083860085, 8.476212218539954738376133982188, 8.864460747184879980214115418809, 9.866982948581862354656193353790, 10.22430495257744996009965708671, 10.72398660732094554317284377299
