| L(s) = 1 | + 3·4-s + 5·16-s + 16·19-s − 4·29-s + 8·31-s + 12·41-s − 49-s + 8·59-s − 4·61-s + 3·64-s + 24·71-s + 48·76-s − 16·79-s − 12·89-s + 20·101-s + 36·109-s − 12·116-s − 22·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 5/4·16-s + 3.67·19-s − 0.742·29-s + 1.43·31-s + 1.87·41-s − 1/7·49-s + 1.04·59-s − 0.512·61-s + 3/8·64-s + 2.84·71-s + 5.50·76-s − 1.80·79-s − 1.27·89-s + 1.99·101-s + 3.44·109-s − 1.11·116-s − 2·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.584011520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.584011520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.515463804908827023448684501502, −9.490983113544726911096892281442, −8.894731256731710977761461520416, −8.232235082359792221382278554871, −7.942708496971895126372494898041, −7.48858141921298755786638729116, −7.15692283948592883349122970358, −7.10458527971566872573636769625, −6.26943173189725074943512818816, −6.10638354132874879162814788058, −5.54589314464760391558465856111, −5.24483514627028430888491547916, −4.74574884356674026826544923732, −4.06139191848469787851416503012, −3.36565902875426010678441343639, −3.21345964266867327788660196480, −2.55451499972035659248183105405, −2.20307336360287510509149123793, −1.23092617630934680890913793523, −0.975673123889942811275946981400,
0.975673123889942811275946981400, 1.23092617630934680890913793523, 2.20307336360287510509149123793, 2.55451499972035659248183105405, 3.21345964266867327788660196480, 3.36565902875426010678441343639, 4.06139191848469787851416503012, 4.74574884356674026826544923732, 5.24483514627028430888491547916, 5.54589314464760391558465856111, 6.10638354132874879162814788058, 6.26943173189725074943512818816, 7.10458527971566872573636769625, 7.15692283948592883349122970358, 7.48858141921298755786638729116, 7.942708496971895126372494898041, 8.232235082359792221382278554871, 8.894731256731710977761461520416, 9.490983113544726911096892281442, 9.515463804908827023448684501502
