| L(s) = 1 | − 9·9-s − 28·11-s + 38·19-s − 28·29-s − 266·31-s + 168·41-s + 661·49-s − 388·59-s − 34·61-s − 1.65e3·71-s − 1.10e3·79-s + 81·81-s + 2.20e3·89-s + 252·99-s + 1.10e3·101-s + 3.68e3·109-s − 2.07e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 0.767·11-s + 0.458·19-s − 0.179·29-s − 1.54·31-s + 0.639·41-s + 1.92·49-s − 0.856·59-s − 0.0713·61-s − 2.76·71-s − 1.57·79-s + 1/9·81-s + 2.62·89-s + 0.255·99-s + 1.08·101-s + 3.23·109-s − 1.55·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.551703346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551703346\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 661 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7710 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 p^{2} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 133 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34742 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 131125 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 39546 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 89818 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 194 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 17 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 175117 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 828 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 453134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 552 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1123410 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1104 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1118065 T^{2} + p^{6} T^{4} \) |
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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
−9.345351384744648606772442839705, −9.236608865731083763355389776747, −8.777134374669988223617351143887, −8.490078953783317211010443185000, −7.66770248606964398152314325281, −7.63425352151050579720158848178, −7.28703502884847903674895974005, −6.72043598227828869950645212363, −6.09493377238305412731079992222, −5.75688026059346262893877801932, −5.46025372065920046688909614585, −4.92064878766796327484429902369, −4.37812333010377834639422300518, −3.98829699584281758309677404845, −3.13964558403526928726489327418, −3.10458170886009886005081542092, −2.22803927767488163653189640446, −1.84938106257661067139180973486, −0.997453533672714072584271601255, −0.33226466449880231217591547039,
0.33226466449880231217591547039, 0.997453533672714072584271601255, 1.84938106257661067139180973486, 2.22803927767488163653189640446, 3.10458170886009886005081542092, 3.13964558403526928726489327418, 3.98829699584281758309677404845, 4.37812333010377834639422300518, 4.92064878766796327484429902369, 5.46025372065920046688909614585, 5.75688026059346262893877801932, 6.09493377238305412731079992222, 6.72043598227828869950645212363, 7.28703502884847903674895974005, 7.63425352151050579720158848178, 7.66770248606964398152314325281, 8.490078953783317211010443185000, 8.777134374669988223617351143887, 9.236608865731083763355389776747, 9.345351384744648606772442839705
