| L(s) = 1 | + 5-s + 9-s + 4·29-s − 4·41-s + 45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 5-s + 9-s + 4·29-s − 4·41-s + 45-s + 2·61-s − 2·89-s − 2·101-s + 2·109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.126749570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126749570\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$ | \( ( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.658324089163570759377829810949, −8.544904331268213053730215793229, −8.076981934288995250842476762173, −7.976548706184446865390362053358, −7.12687349369153388765452661726, −6.78459089254537512914890994565, −6.75185943142755218652918484211, −6.45892390804888911059095724372, −5.84483935684594269645001444564, −5.43929766255881672517468222856, −5.06647583653431159683539426337, −4.78245619302691252748903274452, −4.27086280084149204760307857238, −3.99719055258620313241446925148, −3.19217669200412904210876527403, −3.03745872317234705303229426957, −2.43850170423643638790939319817, −1.85906446390295164884588257946, −1.48812474695699962345333133290, −0.883798214941200687861064139717,
0.883798214941200687861064139717, 1.48812474695699962345333133290, 1.85906446390295164884588257946, 2.43850170423643638790939319817, 3.03745872317234705303229426957, 3.19217669200412904210876527403, 3.99719055258620313241446925148, 4.27086280084149204760307857238, 4.78245619302691252748903274452, 5.06647583653431159683539426337, 5.43929766255881672517468222856, 5.84483935684594269645001444564, 6.45892390804888911059095724372, 6.75185943142755218652918484211, 6.78459089254537512914890994565, 7.12687349369153388765452661726, 7.976548706184446865390362053358, 8.076981934288995250842476762173, 8.544904331268213053730215793229, 8.658324089163570759377829810949
