L(s) = 1 | + 2·4-s + 3·16-s − 2·25-s + 4·64-s − 4·97-s − 4·100-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | + 2·4-s + 3·16-s − 2·25-s + 4·64-s − 4·97-s − 4·100-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.655667825\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.655667825\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + T^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
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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
−10.21096728128016961756596708127, −10.06721808329148398180358714857, −9.375810229740356501027215040802, −9.355268859818653488796930526917, −8.210923552930852750726430637685, −8.201978185270831678360105863139, −7.86999774936843300799674658882, −7.21026740810000805014704578668, −6.92379717397623357864914329454, −6.66798072466537257219409274280, −5.99374002314788415982377879248, −5.72178608052953435384779413066, −5.44367271914378711259131893485, −4.63826198419024545315704109965, −3.85803134120321147141647257358, −3.68341013037308675448901994764, −2.74495422624022727431092388158, −2.62409618595247610835911859261, −1.80435812220385207565973103636, −1.39050397970367248922835682035,
1.39050397970367248922835682035, 1.80435812220385207565973103636, 2.62409618595247610835911859261, 2.74495422624022727431092388158, 3.68341013037308675448901994764, 3.85803134120321147141647257358, 4.63826198419024545315704109965, 5.44367271914378711259131893485, 5.72178608052953435384779413066, 5.99374002314788415982377879248, 6.66798072466537257219409274280, 6.92379717397623357864914329454, 7.21026740810000805014704578668, 7.86999774936843300799674658882, 8.201978185270831678360105863139, 8.210923552930852750726430637685, 9.355268859818653488796930526917, 9.375810229740356501027215040802, 10.06721808329148398180358714857, 10.21096728128016961756596708127