| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 3·9-s + 4·11-s − 2·13-s + 2·15-s + 2·17-s + 9·19-s − 4·21-s + 3·23-s − 5·25-s + 4·27-s − 3·29-s + 31-s + 8·33-s − 2·35-s − 6·37-s − 4·39-s + 6·41-s + 15·43-s + 3·45-s + 47-s + 3·49-s + 4·51-s + 5·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s + 2.06·19-s − 0.872·21-s + 0.625·23-s − 25-s + 0.769·27-s − 0.557·29-s + 0.179·31-s + 1.39·33-s − 0.338·35-s − 0.986·37-s − 0.640·39-s + 0.937·41-s + 2.28·43-s + 0.447·45-s + 0.145·47-s + 3/7·49-s + 0.560·51-s + 0.686·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.410441749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.410441749\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 54 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 56 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 108 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 60 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 186 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 220 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.592031143377707751291020430237, −8.209197905554803173198414239762, −7.63440507596170888808173101697, −7.53123188168722198939407970556, −7.22429220934273708928997619655, −6.89818681012205918023028989733, −6.28203039892872926885278307966, −6.12296435506030564500514203359, −5.52373118515310058780871164410, −5.37480572561691179871554656684, −4.74352431333538609399444049399, −4.32082479291631640997077825386, −3.79083645770729656009810710698, −3.59691775495615444920228810348, −3.06498976679320005147287416152, −2.88340205073481354091870599228, −2.01780281347619080587613979567, −2.01154362202121127135375199172, −1.03569791237203781219273122564, −0.77977158341817037127408942396,
0.77977158341817037127408942396, 1.03569791237203781219273122564, 2.01154362202121127135375199172, 2.01780281347619080587613979567, 2.88340205073481354091870599228, 3.06498976679320005147287416152, 3.59691775495615444920228810348, 3.79083645770729656009810710698, 4.32082479291631640997077825386, 4.74352431333538609399444049399, 5.37480572561691179871554656684, 5.52373118515310058780871164410, 6.12296435506030564500514203359, 6.28203039892872926885278307966, 6.89818681012205918023028989733, 7.22429220934273708928997619655, 7.53123188168722198939407970556, 7.63440507596170888808173101697, 8.209197905554803173198414239762, 8.592031143377707751291020430237
