| L(s) = 1 | − 2·3-s − 2·5-s − 9-s + 2·11-s + 10·13-s + 4·15-s + 10·17-s + 4·19-s − 4·23-s + 3·25-s + 6·27-s + 2·29-s − 12·31-s − 4·33-s − 4·37-s − 20·39-s + 4·41-s − 12·43-s + 2·45-s − 2·47-s − 20·51-s − 16·53-s − 4·55-s − 8·57-s − 4·59-s + 16·61-s − 20·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1/3·9-s + 0.603·11-s + 2.77·13-s + 1.03·15-s + 2.42·17-s + 0.917·19-s − 0.834·23-s + 3/5·25-s + 1.15·27-s + 0.371·29-s − 2.15·31-s − 0.696·33-s − 0.657·37-s − 3.20·39-s + 0.624·41-s − 1.82·43-s + 0.298·45-s − 0.291·47-s − 2.80·51-s − 2.19·53-s − 0.539·55-s − 1.05·57-s − 0.520·59-s + 2.04·61-s − 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926565878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926565878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 T + 49 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 152 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 100 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T - 41 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.516379132552197252517739679872, −8.235952544168133931130814981034, −7.919641727185140360795128122359, −7.75032194148716947323816792832, −6.97023623147530293538794132126, −6.88157680674832652268592355875, −6.23921338505038665102960687093, −6.09025940441812469340437248901, −5.61358791471021521469328509118, −5.50688426457098944014743902909, −5.00595968813989673423295202217, −4.60753845717234667140618259883, −3.79916922149721487993613772481, −3.59940902690060265821013856011, −3.27903334769201808688063940427, −3.27768187429264376530831474516, −1.93637632700613074396246778066, −1.60562502189785482756376910166, −0.818161780749957320772486021336, −0.64622081010090190652660434470,
0.64622081010090190652660434470, 0.818161780749957320772486021336, 1.60562502189785482756376910166, 1.93637632700613074396246778066, 3.27768187429264376530831474516, 3.27903334769201808688063940427, 3.59940902690060265821013856011, 3.79916922149721487993613772481, 4.60753845717234667140618259883, 5.00595968813989673423295202217, 5.50688426457098944014743902909, 5.61358791471021521469328509118, 6.09025940441812469340437248901, 6.23921338505038665102960687093, 6.88157680674832652268592355875, 6.97023623147530293538794132126, 7.75032194148716947323816792832, 7.919641727185140360795128122359, 8.235952544168133931130814981034, 8.516379132552197252517739679872
