| L(s) = 1 | + 5.79e5·13-s + 1.66e8·25-s + 9.36e8·37-s + 2.55e9·49-s − 1.89e10·61-s + 9.10e9·73-s − 3.43e10·97-s − 9.69e11·109-s − 1.04e12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.95e12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 0.432·13-s + 3.40·25-s + 2.21·37-s + 1.29·49-s − 2.87·61-s + 0.514·73-s − 0.405·97-s − 6.03·109-s − 3.66·121-s − 3.88·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.610137569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610137569\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 3329512 p^{2} T^{2} + p^{22} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 182760098 p T^{2} + p^{22} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 522467350822 T^{2} + p^{22} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 144892 T + p^{11} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 67223249355184 T^{2} + p^{22} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 88983405169238 T^{2} + p^{22} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 575810513493746 T^{2} + p^{22} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 20334170784293560 T^{2} + p^{22} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50671712650430462 T^{2} + p^{22} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 234056234 T + p^{11} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 1068466552244168080 T^{2} + p^{22} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 798424803758486 p^{2} T^{2} + p^{22} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 3271983729507535006 T^{2} + p^{22} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 17504183401260332056 T^{2} + p^{22} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 11490778999906753082 T^{2} + p^{22} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4746418690 T + p^{11} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 34823115585889094 p^{2} T^{2} + p^{22} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + \)\(32\!\cdots\!42\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2276138608 T + p^{11} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - \)\(55\!\cdots\!58\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + \)\(25\!\cdots\!34\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - \)\(20\!\cdots\!40\)\( T^{2} + p^{22} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8577858776 T + p^{11} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42100599183979119720642358638, −7.19022643501052993986216714224, −7.07999260514394517884082169076, −6.59178906885854047954454848991, −6.24602876822856683234790494291, −6.14799868457493908953876145566, −6.08404738670553884773886177264, −5.39729576471645020341691242441, −5.05418803863701394658647829156, −5.04086660595106676410157815789, −4.77779521603686452930710972931, −4.20028991204838418898874003354, −4.14051628801460935544221613949, −3.70537768823343671688126265969, −3.51809284488746228753479918881, −2.90350806418822927768179347965, −2.69934131240128406585150105327, −2.51052191800540477778203088651, −2.50850188759147986842950216096, −1.62453119936208181332365605371, −1.24378384222444041996900408157, −1.24376671434439226604305093941, −1.05016890976923157575958995173, −0.49961783497526337756813725688, −0.13621344844775692491313132475,
0.13621344844775692491313132475, 0.49961783497526337756813725688, 1.05016890976923157575958995173, 1.24376671434439226604305093941, 1.24378384222444041996900408157, 1.62453119936208181332365605371, 2.50850188759147986842950216096, 2.51052191800540477778203088651, 2.69934131240128406585150105327, 2.90350806418822927768179347965, 3.51809284488746228753479918881, 3.70537768823343671688126265969, 4.14051628801460935544221613949, 4.20028991204838418898874003354, 4.77779521603686452930710972931, 5.04086660595106676410157815789, 5.05418803863701394658647829156, 5.39729576471645020341691242441, 6.08404738670553884773886177264, 6.14799868457493908953876145566, 6.24602876822856683234790494291, 6.59178906885854047954454848991, 7.07999260514394517884082169076, 7.19022643501052993986216714224, 7.42100599183979119720642358638
