| L(s) = 1 | − 26·3-s + 250·5-s − 686·7-s + 729·9-s − 2.52e3·11-s + 2.77e3·13-s − 6.50e3·15-s − 754·17-s + 1.78e4·21-s + 4.68e4·25-s − 3.92e4·27-s + 4.58e4·29-s + 6.55e4·33-s − 1.71e5·35-s − 7.21e4·39-s + 1.82e5·45-s + 1.75e5·47-s + 3.52e5·49-s + 1.96e4·51-s − 6.30e5·55-s − 5.00e5·63-s + 6.93e5·65-s − 6.04e4·71-s − 1.00e6·73-s − 1.21e6·75-s + 1.73e6·77-s + 9.30e5·79-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 2·5-s − 2·7-s + 9-s − 1.89·11-s + 1.26·13-s − 1.92·15-s − 0.153·17-s + 1.92·21-s + 3·25-s − 1.99·27-s + 1.88·29-s + 1.82·33-s − 4·35-s − 1.21·39-s + 2·45-s + 1.69·47-s + 3·49-s + 0.147·51-s − 3.78·55-s − 2·63-s + 2.52·65-s − 0.168·71-s − 2.59·73-s − 2.88·75-s + 3.78·77-s + 1.88·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.754946437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754946437\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 26 T - 53 T^{2} + 26 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2522 T + 4588923 T^{2} + 2522 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2774 T + 2868267 T^{2} - 2774 p^{6} T^{3} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 754 T - 23569053 T^{2} + 754 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 45862 T + 1508499723 T^{2} - 45862 p^{6} T^{3} + p^{12} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 175646 T + 20072301987 T^{2} - 175646 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 30238 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 504254 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 930382 T + 622523210403 T^{2} - 930382 p^{6} T^{3} + p^{12} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1141306 T + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 897874 T - 26794285053 T^{2} + 897874 p^{6} T^{3} + p^{12} T^{4} \) |
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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
−12.28927409277170436899588470922, −12.08144278072829382490609643399, −10.75413729838581105703947463171, −10.69663600278646748166582821141, −10.34367231862457689870072695450, −9.818893055292104097457618763744, −9.342537729892185113098139463866, −8.923699758192344326639795106845, −8.046707622495891477487985534756, −7.21539669112098944891529848022, −6.48055902157357949455402720543, −6.37426779517630479338076868742, −5.60670212444136970241070918359, −5.51859554080795620219388006001, −4.59775787637310996590395941444, −3.58137146338735302595931685951, −2.75079218706973605086278694846, −2.29398083514279879025235957470, −1.18186335071570825750848028385, −0.46860811097315221016475692522,
0.46860811097315221016475692522, 1.18186335071570825750848028385, 2.29398083514279879025235957470, 2.75079218706973605086278694846, 3.58137146338735302595931685951, 4.59775787637310996590395941444, 5.51859554080795620219388006001, 5.60670212444136970241070918359, 6.37426779517630479338076868742, 6.48055902157357949455402720543, 7.21539669112098944891529848022, 8.046707622495891477487985534756, 8.923699758192344326639795106845, 9.342537729892185113098139463866, 9.818893055292104097457618763744, 10.34367231862457689870072695450, 10.69663600278646748166582821141, 10.75413729838581105703947463171, 12.08144278072829382490609643399, 12.28927409277170436899588470922
