L(s) = 1 | − 2-s + 4-s − 4·5-s − 8-s + 4·10-s + 4·13-s + 16-s + 12·17-s − 4·20-s + 2·25-s − 4·26-s + 4·29-s − 32-s − 12·34-s + 20·37-s + 4·40-s − 20·41-s − 14·49-s − 2·50-s + 4·52-s + 20·53-s − 4·58-s + 28·61-s + 64-s − 16·65-s + 12·68-s − 12·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1.26·10-s + 1.10·13-s + 1/4·16-s + 2.91·17-s − 0.894·20-s + 2/5·25-s − 0.784·26-s + 0.742·29-s − 0.176·32-s − 2.05·34-s + 3.28·37-s + 0.632·40-s − 3.12·41-s − 2·49-s − 0.282·50-s + 0.554·52-s + 2.74·53-s − 0.525·58-s + 3.58·61-s + 1/8·64-s − 1.98·65-s + 1.45·68-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.178503068\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178503068\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.057224083383321383561004935430, −7.87114970877127535090362209983, −7.54188937600423423101808663065, −6.93381933813641935760483898363, −6.59739773494044819386161896201, −5.88087715977795198376928556594, −5.60441811878021212455097225817, −5.02382748416547556621669886282, −4.27229588770101457625379469521, −3.86301997176052042051145556459, −3.33052988139656335920774502455, −3.17802027729570401593965112053, −2.15606001082909168122252072981, −1.15871174429474862980802238748, −0.69408070740984200675842868720,
0.69408070740984200675842868720, 1.15871174429474862980802238748, 2.15606001082909168122252072981, 3.17802027729570401593965112053, 3.33052988139656335920774502455, 3.86301997176052042051145556459, 4.27229588770101457625379469521, 5.02382748416547556621669886282, 5.60441811878021212455097225817, 5.88087715977795198376928556594, 6.59739773494044819386161896201, 6.93381933813641935760483898363, 7.54188937600423423101808663065, 7.87114970877127535090362209983, 8.057224083383321383561004935430