| L(s) = 1 | − 2·5-s − 6·9-s − 8·11-s − 4·13-s + 4·17-s + 2·19-s + 3·25-s − 4·29-s − 4·37-s + 4·41-s + 12·45-s − 16·47-s − 6·49-s − 20·53-s + 16·55-s + 8·59-s − 12·61-s + 8·65-s + 20·73-s + 27·81-s + 16·83-s − 8·85-s − 12·89-s − 4·95-s + 20·97-s + 48·99-s − 28·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2·9-s − 2.41·11-s − 1.10·13-s + 0.970·17-s + 0.458·19-s + 3/5·25-s − 0.742·29-s − 0.657·37-s + 0.624·41-s + 1.78·45-s − 2.33·47-s − 6/7·49-s − 2.74·53-s + 2.15·55-s + 1.04·59-s − 1.53·61-s + 0.992·65-s + 2.34·73-s + 3·81-s + 1.75·83-s − 0.867·85-s − 1.27·89-s − 0.410·95-s + 2.03·97-s + 4.82·99-s − 2.78·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5766182620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5766182620\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_4$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20 T + 286 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 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
−7.973962775328381465141514081159, −7.958073147759736130376427671892, −7.67269729226539941038386971183, −7.51894737462940752441559642158, −6.80128232945254203188204634585, −6.52024548900929494759496879132, −6.03168971323719976459673884359, −5.65003813319815168918465373925, −5.25669713954614332723992018648, −5.12280770126786225258143289036, −4.77557745346972188319980637331, −4.44226161935373760416553399190, −3.42904317678032786222453287407, −3.41925405995312941702367004485, −2.95282572813183274592520475074, −2.86876422178646788629450422299, −2.01722850055231064384495222783, −1.91997308987285852098995476958, −0.61347622094169007204538530197, −0.30687705162094048799156643804,
0.30687705162094048799156643804, 0.61347622094169007204538530197, 1.91997308987285852098995476958, 2.01722850055231064384495222783, 2.86876422178646788629450422299, 2.95282572813183274592520475074, 3.41925405995312941702367004485, 3.42904317678032786222453287407, 4.44226161935373760416553399190, 4.77557745346972188319980637331, 5.12280770126786225258143289036, 5.25669713954614332723992018648, 5.65003813319815168918465373925, 6.03168971323719976459673884359, 6.52024548900929494759496879132, 6.80128232945254203188204634585, 7.51894737462940752441559642158, 7.67269729226539941038386971183, 7.958073147759736130376427671892, 7.973962775328381465141514081159
