| L(s) = 1 | + 2-s − 5-s − 7-s − 8-s − 10-s − 11-s − 2·13-s − 14-s − 16-s − 6·17-s + 4·19-s − 22-s − 6·23-s + 5·25-s − 2·26-s − 6·29-s + 11·31-s − 6·34-s + 35-s − 4·37-s + 4·38-s + 40-s − 24·41-s − 16·43-s − 6·46-s − 8·47-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.213·22-s − 1.25·23-s + 25-s − 0.392·26-s − 1.11·29-s + 1.97·31-s − 1.02·34-s + 0.169·35-s − 0.657·37-s + 0.648·38-s + 0.158·40-s − 3.74·41-s − 2.43·43-s − 0.884·46-s − 1.16·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3360870702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3360870702\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−9.588069811823785900206029230073, −9.060231577130799483457904630216, −8.900491691912884862722186103866, −8.179017573224966050743454760555, −8.153723819119847571039558326512, −7.60593575585746626315253757533, −7.07555285773896070643310115943, −6.60312244434609747972060436780, −6.42214236536942418995852031888, −6.07284172462992005762036370668, −5.20211248999822028088127015889, −4.88089532561439488080522064357, −4.82933208391977207310395101827, −4.24642601059657227553588949510, −3.42434289499260953192927895241, −3.29837441410369453645198077481, −2.92472527366035284033214803339, −1.91472445197289515713796855005, −1.67336930657493277031274457375, −0.18706138453915100679539187223,
0.18706138453915100679539187223, 1.67336930657493277031274457375, 1.91472445197289515713796855005, 2.92472527366035284033214803339, 3.29837441410369453645198077481, 3.42434289499260953192927895241, 4.24642601059657227553588949510, 4.82933208391977207310395101827, 4.88089532561439488080522064357, 5.20211248999822028088127015889, 6.07284172462992005762036370668, 6.42214236536942418995852031888, 6.60312244434609747972060436780, 7.07555285773896070643310115943, 7.60593575585746626315253757533, 8.153723819119847571039558326512, 8.179017573224966050743454760555, 8.900491691912884862722186103866, 9.060231577130799483457904630216, 9.588069811823785900206029230073
