| L(s) = 1 | + 3·7-s − 13-s + 25-s − 2·37-s + 5·49-s + 61-s + 3·67-s + 2·73-s − 3·79-s − 3·91-s − 97-s − 3·103-s − 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 3·175-s + 179-s + ⋯ |
| L(s) = 1 | + 3·7-s − 13-s + 25-s − 2·37-s + 5·49-s + 61-s + 3·67-s + 2·73-s − 3·79-s − 3·91-s − 97-s − 3·103-s − 4·109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 3·175-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.506831964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.506831964\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.996510212850216340216091384203, −9.791902615239521127313433528469, −8.938286671564841834798194742013, −8.921861179008962435575415001843, −8.222003787667362122764573021420, −8.106568627095995396899443865739, −7.84065385959192551477496001007, −7.16106758434397464540285792914, −6.92202543814077441186939196323, −6.54017162726213376600485443005, −5.49121248257260812318924678306, −5.33514938889448420725420702813, −5.04319538511313213439944940436, −4.74068328645185722852966382623, −3.93033597963368283782241038190, −3.88679375911641423821655249687, −2.59408595753712814717891629242, −2.48166398398396490622656458135, −1.58358352662493684964319040722, −1.30334691944177848365706450745,
1.30334691944177848365706450745, 1.58358352662493684964319040722, 2.48166398398396490622656458135, 2.59408595753712814717891629242, 3.88679375911641423821655249687, 3.93033597963368283782241038190, 4.74068328645185722852966382623, 5.04319538511313213439944940436, 5.33514938889448420725420702813, 5.49121248257260812318924678306, 6.54017162726213376600485443005, 6.92202543814077441186939196323, 7.16106758434397464540285792914, 7.84065385959192551477496001007, 8.106568627095995396899443865739, 8.222003787667362122764573021420, 8.921861179008962435575415001843, 8.938286671564841834798194742013, 9.791902615239521127313433528469, 9.996510212850216340216091384203
