| L(s) = 1 | + 9·3-s + 5·5-s + 27·9-s − 55·11-s + 138·13-s + 45·15-s + 113·17-s − 126·19-s + 102·23-s − 162·29-s + 176·31-s − 495·33-s − 254·37-s + 1.24e3·39-s + 368·41-s − 460·43-s + 135·45-s − 187·47-s + 1.01e3·51-s + 488·53-s − 275·55-s − 1.13e3·57-s + 388·59-s − 728·61-s + 690·65-s + 96·67-s + 918·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 9-s − 1.50·11-s + 2.94·13-s + 0.774·15-s + 1.61·17-s − 1.52·19-s + 0.924·23-s − 1.03·29-s + 1.01·31-s − 2.61·33-s − 1.12·37-s + 5.09·39-s + 1.40·41-s − 1.63·43-s + 0.447·45-s − 0.580·47-s + 2.79·51-s + 1.26·53-s − 0.674·55-s − 2.63·57-s + 0.856·59-s − 1.52·61-s + 1.31·65-s + 0.175·67-s + 1.60·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.905333848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.905333848\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 p T + 14 p^{2} T^{2} + 5 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 69 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 113 T + 7856 T^{2} - 113 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 126 T + 9017 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 102 T - 1763 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 81 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 176 T + 1185 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 184 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 230 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 187 T - 68854 T^{2} + 187 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 488 T + 89267 T^{2} - 488 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 388 T - 54835 T^{2} - 388 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 728 T + 303003 T^{2} + 728 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 96 T - 291547 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 994 T + 599019 T^{2} + 994 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 337 T - 379470 T^{2} + 337 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 188 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 884 T + 76487 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 451 T + p^{3} T^{2} )^{2} \) |
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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
−10.14261014737247860034337704051, −9.155346717024231363183482540887, −8.806426551356429157505819064910, −8.554794434252869602176418206440, −8.347723674111833645863426769508, −7.992781436130152985218458996325, −7.40966468730964041393267369556, −7.09367764105970041244877528780, −6.15949052910746895381586711686, −6.11869213216331569190509553483, −5.61310079608461541274249888184, −5.03830000273657329676244192623, −4.42449735111409688206281377170, −3.63562808921415172005381170068, −3.53075144175911997648249950760, −2.96781940842051868869340587288, −2.57644014885629904215803856403, −1.82262742748288898904810279430, −1.40139370556516881180211607106, −0.58065679539542850787255512440,
0.58065679539542850787255512440, 1.40139370556516881180211607106, 1.82262742748288898904810279430, 2.57644014885629904215803856403, 2.96781940842051868869340587288, 3.53075144175911997648249950760, 3.63562808921415172005381170068, 4.42449735111409688206281377170, 5.03830000273657329676244192623, 5.61310079608461541274249888184, 6.11869213216331569190509553483, 6.15949052910746895381586711686, 7.09367764105970041244877528780, 7.40966468730964041393267369556, 7.992781436130152985218458996325, 8.347723674111833645863426769508, 8.554794434252869602176418206440, 8.806426551356429157505819064910, 9.155346717024231363183482540887, 10.14261014737247860034337704051
