| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 9-s + 14-s + 16-s + 9·17-s + 18-s − 15·23-s + 2·25-s + 28-s − 2·31-s + 32-s + 9·34-s + 36-s + 3·41-s − 15·46-s + 15·47-s + 49-s + 2·50-s + 56-s − 2·62-s + 63-s + 64-s + 9·68-s − 9·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.267·14-s + 1/4·16-s + 2.18·17-s + 0.235·18-s − 3.12·23-s + 2/5·25-s + 0.188·28-s − 0.359·31-s + 0.176·32-s + 1.54·34-s + 1/6·36-s + 0.468·41-s − 2.21·46-s + 2.18·47-s + 1/7·49-s + 0.282·50-s + 0.133·56-s − 0.254·62-s + 0.125·63-s + 1/8·64-s + 1.09·68-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.220315274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.220315274\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−10.19976140167580211847522753618, −9.983276814852255691104147335999, −9.228456935030750636844611953843, −8.529355233718924661459974058399, −7.973697248311446278928268086973, −7.51329354649218487872713796635, −7.19745843863754257057989241632, −6.14780891076497166196428672263, −5.81568324292222221465354894664, −5.42132046764882156415928786637, −4.43834535538038720537561674323, −4.03608470320456923138623011939, −3.32985802399066058978527190800, −2.39327020305493056227305745456, −1.42390401050368327152943717813,
1.42390401050368327152943717813, 2.39327020305493056227305745456, 3.32985802399066058978527190800, 4.03608470320456923138623011939, 4.43834535538038720537561674323, 5.42132046764882156415928786637, 5.81568324292222221465354894664, 6.14780891076497166196428672263, 7.19745843863754257057989241632, 7.51329354649218487872713796635, 7.973697248311446278928268086973, 8.529355233718924661459974058399, 9.228456935030750636844611953843, 9.983276814852255691104147335999, 10.19976140167580211847522753618
