| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s + 2·9-s − 11-s + 3·13-s + 3·14-s + 16-s + 2·18-s − 22-s + 25-s + 3·26-s + 3·28-s − 15·31-s + 32-s + 2·36-s + 19·43-s − 44-s + 6·47-s + 2·49-s + 50-s + 3·52-s + 3·56-s − 9·61-s − 15·62-s + 6·63-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s + 2/3·9-s − 0.301·11-s + 0.832·13-s + 0.801·14-s + 1/4·16-s + 0.471·18-s − 0.213·22-s + 1/5·25-s + 0.588·26-s + 0.566·28-s − 2.69·31-s + 0.176·32-s + 1/3·36-s + 2.89·43-s − 0.150·44-s + 0.875·47-s + 2/7·49-s + 0.141·50-s + 0.416·52-s + 0.400·56-s − 1.15·61-s − 1.90·62-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.192585091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192585091\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.250455946136333976462231732846, −8.778152995613561187456367550962, −8.195168725218209714344742497234, −7.66758002575102483759024157596, −7.34304601647805467997182858014, −6.90459388794169053719779810550, −6.11540564075263288406795777270, −5.64671258666161510936091971914, −5.27763568977483214284301647846, −4.59000146859169662112078614756, −4.05387356401999072909146552679, −3.68070032399654671761606844650, −2.71102238748866048869998099594, −1.96485111121689795653963688023, −1.23472302579262159122189030442,
1.23472302579262159122189030442, 1.96485111121689795653963688023, 2.71102238748866048869998099594, 3.68070032399654671761606844650, 4.05387356401999072909146552679, 4.59000146859169662112078614756, 5.27763568977483214284301647846, 5.64671258666161510936091971914, 6.11540564075263288406795777270, 6.90459388794169053719779810550, 7.34304601647805467997182858014, 7.66758002575102483759024157596, 8.195168725218209714344742497234, 8.778152995613561187456367550962, 9.250455946136333976462231732846
