| L(s) = 1 | − 2·3-s + 6·4-s − 14·5-s − 15·9-s − 12·12-s + 28·15-s + 20·16-s − 84·20-s − 18·23-s + 97·25-s + 50·27-s + 98·31-s − 90·36-s + 34·37-s + 210·45-s + 64·47-s − 40·48-s + 48·49-s + 32·53-s − 142·59-s + 168·60-s + 24·64-s − 62·67-s + 36·69-s − 146·71-s − 194·75-s − 280·80-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 3/2·4-s − 2.79·5-s − 5/3·9-s − 12-s + 1.86·15-s + 5/4·16-s − 4.19·20-s − 0.782·23-s + 3.87·25-s + 1.85·27-s + 3.16·31-s − 5/2·36-s + 0.918·37-s + 14/3·45-s + 1.36·47-s − 5/6·48-s + 0.979·49-s + 0.603·53-s − 2.40·59-s + 14/5·60-s + 3/8·64-s − 0.925·67-s + 0.521·69-s − 2.05·71-s − 2.58·75-s − 7/2·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6912191406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6912191406\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 3 p T^{2} + p^{4} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 560 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1170 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 49 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 1520 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 71 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 7314 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9090 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 12160 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12528 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
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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
−13.99809295019528776467839866676, −12.31327984556556409245766645905, −12.12443687813786556412911975660, −11.76035804095524832777040055369, −11.71865240722195021047500115730, −11.01226689603628692536461240505, −10.80684206659331420926479415779, −10.18057905564263407761932766230, −9.006010627647634252042984412425, −8.206277608333710645597970272249, −8.191179188687478119723275874521, −7.47459721215388679673598390824, −7.04341679631888682405936964416, −6.05639903470724050921112103362, −6.01535174134138459993401858676, −4.67570851087674830887363985095, −4.16483026056663427569047947614, −3.09449157917602735179416906144, −2.74924310659488094546090791855, −0.59044975922244403692719338873,
0.59044975922244403692719338873, 2.74924310659488094546090791855, 3.09449157917602735179416906144, 4.16483026056663427569047947614, 4.67570851087674830887363985095, 6.01535174134138459993401858676, 6.05639903470724050921112103362, 7.04341679631888682405936964416, 7.47459721215388679673598390824, 8.191179188687478119723275874521, 8.206277608333710645597970272249, 9.006010627647634252042984412425, 10.18057905564263407761932766230, 10.80684206659331420926479415779, 11.01226689603628692536461240505, 11.71865240722195021047500115730, 11.76035804095524832777040055369, 12.12443687813786556412911975660, 12.31327984556556409245766645905, 13.99809295019528776467839866676
