| L(s) = 1 | − 2·5-s − 2·7-s + 2·11-s − 6·13-s − 4·17-s + 6·23-s + 3·25-s − 10·29-s + 4·31-s + 4·35-s − 4·37-s − 2·41-s − 6·43-s + 4·47-s + 2·49-s + 4·53-s − 4·55-s − 10·59-s − 6·61-s + 12·65-s − 16·67-s − 22·71-s + 18·73-s − 4·77-s + 16·79-s + 6·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 0.603·11-s − 1.66·13-s − 0.970·17-s + 1.25·23-s + 3/5·25-s − 1.85·29-s + 0.718·31-s + 0.676·35-s − 0.657·37-s − 0.312·41-s − 0.914·43-s + 0.583·47-s + 2/7·49-s + 0.549·53-s − 0.539·55-s − 1.30·59-s − 0.768·61-s + 1.48·65-s − 1.95·67-s − 2.61·71-s + 2.10·73-s − 0.455·77-s + 1.80·79-s + 0.658·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74649600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74649600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5666345583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5666345583\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 97 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 209 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.70632833151679471969597638770, −7.60256775712204749033977593540, −7.16520258915060834292562503331, −7.09225360929528379305254187194, −6.59701678321997336210569977864, −6.38218195033057287521490459607, −5.91127698403205633915644990705, −5.52421882812377620253060636273, −4.92971684757104726137020008901, −4.92169686693324050406195078916, −4.36946934110658168733782454783, −4.17030716142106653073355952767, −3.52192983816776722941219935832, −3.39799068471112309821591226189, −2.81635610571740390589594413841, −2.64413322820254369112246292157, −1.86506103161801755090970266778, −1.67874037806186606832636637034, −0.76848151133726351394786359748, −0.22658921592113570937878268213,
0.22658921592113570937878268213, 0.76848151133726351394786359748, 1.67874037806186606832636637034, 1.86506103161801755090970266778, 2.64413322820254369112246292157, 2.81635610571740390589594413841, 3.39799068471112309821591226189, 3.52192983816776722941219935832, 4.17030716142106653073355952767, 4.36946934110658168733782454783, 4.92169686693324050406195078916, 4.92971684757104726137020008901, 5.52421882812377620253060636273, 5.91127698403205633915644990705, 6.38218195033057287521490459607, 6.59701678321997336210569977864, 7.09225360929528379305254187194, 7.16520258915060834292562503331, 7.60256775712204749033977593540, 7.70632833151679471969597638770
