| L(s) = 1 | + 2·3-s + 4-s − 2·7-s + 3·9-s + 2·12-s − 2·13-s − 4·21-s + 2·25-s + 4·27-s − 2·28-s + 2·31-s + 3·36-s − 2·37-s − 4·39-s + 3·49-s − 2·52-s + 2·61-s − 6·63-s − 64-s − 2·67-s + 4·75-s + 5·81-s − 4·84-s + 4·91-s + 4·93-s + 2·97-s + 2·100-s + ⋯ |
| L(s) = 1 | + 2·3-s + 4-s − 2·7-s + 3·9-s + 2·12-s − 2·13-s − 4·21-s + 2·25-s + 4·27-s − 2·28-s + 2·31-s + 3·36-s − 2·37-s − 4·39-s + 3·49-s − 2·52-s + 2·61-s − 6·63-s − 64-s − 2·67-s + 4·75-s + 5·81-s − 4·84-s + 4·91-s + 4·93-s + 2·97-s + 2·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1979649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1979649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.255918976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.255918976\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 67 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08158281416385301254211949447, −9.589698309942117115047265663795, −8.966850844614646388332724962895, −8.894606665426277292739774159957, −8.527617317553105533227666464322, −7.87746537547716892781578287065, −7.35975736171364564088824467347, −7.11456880398689850938106404332, −6.97688617569973215845580586711, −6.45119629043731757561653958376, −6.21925264726170474147738184021, −5.25684651220761681199517314856, −4.73833252995485460050379003337, −4.43477596012746826551058310764, −3.58917456403635046295167939687, −3.29476143755037510241597064041, −2.86704674829045291873355193560, −2.37801297364468766931040970657, −2.31253313153691854372057574601, −1.17905194556107831478681965805,
1.17905194556107831478681965805, 2.31253313153691854372057574601, 2.37801297364468766931040970657, 2.86704674829045291873355193560, 3.29476143755037510241597064041, 3.58917456403635046295167939687, 4.43477596012746826551058310764, 4.73833252995485460050379003337, 5.25684651220761681199517314856, 6.21925264726170474147738184021, 6.45119629043731757561653958376, 6.97688617569973215845580586711, 7.11456880398689850938106404332, 7.35975736171364564088824467347, 7.87746537547716892781578287065, 8.527617317553105533227666464322, 8.894606665426277292739774159957, 8.966850844614646388332724962895, 9.589698309942117115047265663795, 10.08158281416385301254211949447
