| L(s) = 1 | − 2·3-s + 3·9-s − 8·19-s − 2·25-s − 4·27-s + 4·29-s − 12·37-s − 16·47-s + 12·53-s + 16·57-s − 24·59-s + 4·75-s + 5·81-s + 8·83-s − 8·87-s − 32·103-s + 20·109-s + 24·111-s + 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 32·141-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.83·19-s − 2/5·25-s − 0.769·27-s + 0.742·29-s − 1.97·37-s − 2.33·47-s + 1.64·53-s + 2.11·57-s − 3.12·59-s + 0.461·75-s + 5/9·81-s + 0.878·83-s − 0.857·87-s − 3.15·103-s + 1.91·109-s + 2.27·111-s + 3.38·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.69·141-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 170 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 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
−8.045317270904407179168127782671, −7.76933996509458185281986748980, −7.36004113950989869142129919925, −6.81189191556648109007293423016, −6.56408406591183218778267573967, −6.47078291774047370127331727952, −5.88966250960073648159509475387, −5.66977564535409528524101846806, −5.05800274523150911462167768773, −4.92072100236572330145899049945, −4.32074649648640119590403185364, −4.27323572528866642813948923203, −3.41255164551122973296203610149, −3.38355135209059472188426201663, −2.55086394630257134694884034574, −2.04320818966759348032664634680, −1.63211381272935732782874501643, −1.07444200024290472536339495364, 0, 0,
1.07444200024290472536339495364, 1.63211381272935732782874501643, 2.04320818966759348032664634680, 2.55086394630257134694884034574, 3.38355135209059472188426201663, 3.41255164551122973296203610149, 4.27323572528866642813948923203, 4.32074649648640119590403185364, 4.92072100236572330145899049945, 5.05800274523150911462167768773, 5.66977564535409528524101846806, 5.88966250960073648159509475387, 6.47078291774047370127331727952, 6.56408406591183218778267573967, 6.81189191556648109007293423016, 7.36004113950989869142129919925, 7.76933996509458185281986748980, 8.045317270904407179168127782671
