| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s − 3·9-s − 4·13-s − 4·15-s + 10·19-s + 4·21-s + 2·23-s − 2·25-s − 14·27-s + 6·29-s + 18·31-s − 4·35-s + 2·37-s − 8·39-s + 4·43-s + 6·45-s − 2·47-s + 3·49-s + 18·53-s + 20·57-s + 12·59-s − 6·63-s + 8·65-s + 2·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s − 9-s − 1.10·13-s − 1.03·15-s + 2.29·19-s + 0.872·21-s + 0.417·23-s − 2/5·25-s − 2.69·27-s + 1.11·29-s + 3.23·31-s − 0.676·35-s + 0.328·37-s − 1.28·39-s + 0.609·43-s + 0.894·45-s − 0.291·47-s + 3/7·49-s + 2.47·53-s + 2.64·57-s + 1.56·59-s − 0.755·63-s + 0.992·65-s + 0.244·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6635776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6635776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.484033922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.484033922\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 201 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 178 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 158 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.944380108876022801650734435709, −8.489015612259476422521397059972, −8.212686644770663944643367150137, −8.173820523511472007915594724913, −7.57936137611453009807074963420, −7.48985068612798017467480657882, −6.91424316748024625755178213046, −6.52903438750615800839683333880, −5.87922716924499820150219554383, −5.51078413792474492063838923012, −5.01657147455405927444939910530, −4.89403119134647697712969323441, −4.11246691187800336651435789394, −3.89281773443994275913875936624, −3.19631257719102555392379570398, −2.95600226613732462711913907315, −2.41000161826508758512812095433, −2.28626078453958104937122590742, −1.04371521834324393162423492411, −0.68626938944931894447992867781,
0.68626938944931894447992867781, 1.04371521834324393162423492411, 2.28626078453958104937122590742, 2.41000161826508758512812095433, 2.95600226613732462711913907315, 3.19631257719102555392379570398, 3.89281773443994275913875936624, 4.11246691187800336651435789394, 4.89403119134647697712969323441, 5.01657147455405927444939910530, 5.51078413792474492063838923012, 5.87922716924499820150219554383, 6.52903438750615800839683333880, 6.91424316748024625755178213046, 7.48985068612798017467480657882, 7.57936137611453009807074963420, 8.173820523511472007915594724913, 8.212686644770663944643367150137, 8.489015612259476422521397059972, 8.944380108876022801650734435709
