| L(s) = 1 | + 2·2-s − 5·5-s + 22·7-s − 8·8-s − 10·10-s − 12·11-s − 38·13-s + 44·14-s − 16·16-s + 210·17-s − 314·19-s − 24·22-s − 117·23-s − 76·26-s + 66·29-s + 25·31-s + 420·34-s − 110·35-s + 628·37-s − 628·38-s + 40·40-s − 504·41-s − 380·43-s − 234·46-s − 252·47-s + 343·49-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s + 1.18·7-s − 0.353·8-s − 0.316·10-s − 0.328·11-s − 0.810·13-s + 0.839·14-s − 1/4·16-s + 2.99·17-s − 3.79·19-s − 0.232·22-s − 1.06·23-s − 0.573·26-s + 0.422·29-s + 0.144·31-s + 2.11·34-s − 0.531·35-s + 2.79·37-s − 2.68·38-s + 0.158·40-s − 1.91·41-s − 1.34·43-s − 0.750·46-s − 0.782·47-s + 49-s − 0.0155·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.206791499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206791499\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 22 T + 141 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 38 T - 753 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 105 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 157 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 117 T + 1522 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 66 T - 20033 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 25 T - 29166 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 314 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 504 T + 185095 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 380 T + 64893 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 252 T - 40319 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 318 T - 104255 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 293 T - 141132 T^{2} + 293 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 322 T - 197079 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 917 T + 347850 T^{2} + 917 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 309 T - 476306 T^{2} - 309 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1272 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1328 T + 850911 T^{2} + 1328 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.32310079850714570134728317618, −9.640718110465372155674447483974, −9.402801795636124185785972389093, −8.381840690799742040192029065252, −8.293600106156569565133511348856, −7.938781105939453653378450494171, −7.87710416146303603393738228835, −6.82207966331381536559996766244, −6.68837252113636130259731759208, −5.98568793503900486178271469123, −5.55392083307626242918396739615, −5.17848193674032324778745052750, −4.56634914606467185600025818664, −4.17988434281970642206530107805, −3.97282039075007971402284918646, −3.02315604666529819145753193727, −2.66115128845323111753177665246, −1.84761429409608250972532611833, −1.37078257396740041879656305229, −0.25570659414586468780190549490,
0.25570659414586468780190549490, 1.37078257396740041879656305229, 1.84761429409608250972532611833, 2.66115128845323111753177665246, 3.02315604666529819145753193727, 3.97282039075007971402284918646, 4.17988434281970642206530107805, 4.56634914606467185600025818664, 5.17848193674032324778745052750, 5.55392083307626242918396739615, 5.98568793503900486178271469123, 6.68837252113636130259731759208, 6.82207966331381536559996766244, 7.87710416146303603393738228835, 7.938781105939453653378450494171, 8.293600106156569565133511348856, 8.381840690799742040192029065252, 9.402801795636124185785972389093, 9.640718110465372155674447483974, 10.32310079850714570134728317618
