| L(s) = 1 | + 14·3-s − 26·4-s − 38·5-s − 339·9-s − 364·12-s − 532·15-s − 348·16-s + 988·20-s + 6.14e3·23-s − 5.16e3·25-s − 8.83e3·27-s − 3.16e3·31-s + 8.81e3·36-s − 1.82e4·37-s + 1.28e4·45-s − 3.32e4·47-s − 4.87e3·48-s − 3.11e4·49-s − 3.25e4·53-s + 2.90e4·59-s + 1.38e4·60-s + 3.56e4·64-s − 2.12e4·67-s + 8.59e4·69-s + 6.20e4·71-s − 7.23e4·75-s + 1.32e4·80-s + ⋯ |
| L(s) = 1 | + 0.898·3-s − 0.812·4-s − 0.679·5-s − 1.39·9-s − 0.729·12-s − 0.610·15-s − 0.339·16-s + 0.552·20-s + 2.42·23-s − 1.65·25-s − 2.33·27-s − 0.590·31-s + 1.13·36-s − 2.19·37-s + 0.948·45-s − 2.19·47-s − 0.305·48-s − 1.85·49-s − 1.59·53-s + 1.08·59-s + 0.496·60-s + 1.08·64-s − 0.578·67-s + 2.17·69-s + 1.46·71-s − 1.48·75-s + 0.231·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 13 p T^{2} + p^{10} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 7 T + p^{5} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 19 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 31182 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 195386 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 489694 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 23970 p T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3071 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40779098 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 51 p T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 9145 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 71336686 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 95089014 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 16636 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 16266 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14505 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 1628480154 T^{2} + p^{10} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10635 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 31045 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3011641938 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 857839330 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 683252486 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 109481 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13615 T + p^{5} T^{2} )^{2} \) |
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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
−12.36924908140251050916223362137, −11.51115705876298014244318773416, −11.31926463724405542552456793463, −10.98561933365643465774322488000, −9.815323469125278900139875501848, −9.586847145289665824475545738332, −8.784951963728896265176800894510, −8.674097786432749088081036344780, −8.118781378016770780858446352580, −7.55403761872482516088817223428, −6.82662712442880588104621251740, −6.08051047133140866588915963105, −5.06356317718718722950276249484, −4.98697938009708614054218653179, −3.65770743145247350861541347055, −3.46437978352597760446699600659, −2.64382532003488969635477437109, −1.62353433597525133202022977064, 0, 0,
1.62353433597525133202022977064, 2.64382532003488969635477437109, 3.46437978352597760446699600659, 3.65770743145247350861541347055, 4.98697938009708614054218653179, 5.06356317718718722950276249484, 6.08051047133140866588915963105, 6.82662712442880588104621251740, 7.55403761872482516088817223428, 8.118781378016770780858446352580, 8.674097786432749088081036344780, 8.784951963728896265176800894510, 9.586847145289665824475545738332, 9.815323469125278900139875501848, 10.98561933365643465774322488000, 11.31926463724405542552456793463, 11.51115705876298014244318773416, 12.36924908140251050916223362137
